#72255895

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14885
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(10,15)(16,19)(17,23)(18,27)(20,36)(24,39)(31,34)(33,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19,22,16)(12,23)(13,27)(14,15)(17,26)(18,29)(20,35,36,30)(21,31,28,34)(24,32,39,37)(25,33)(38,40);(6,8)(9,11)(10,16)(12,24)(13,28)(14,22)(15,19)(17,32)(18,34)(20,25)(21,29)(23,37)(26,39)(27,31)(30,33)(35,38)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20,22,36)(13,25)(14,26)(15,23)(16,30,19,35)(18,33)(21,24,28,39)(27,38)(29,40)(31,32,34,37)

Polycyclic presentation:

2^10|b^2=h,d^2=hj,e^2=ij,b^a=bi,c^a=cf,c^b=cg,d^a=dh,d^b=dhij,d^c=de,e^b=ei,e^c=eij,f^d=fij,g^b=gij,g^d=gj,h^c=hij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255950, #72256124, #72256271
maximal normal subgroup of	#72255950, #72256124, #72256271

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.8*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);