#72250592

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1874
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#24281	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=g,d^2=eghj,e^2=ij,g^2=j,i^2=j,b^a=bf,c^b=cfgij,d^b=de,d^c=degj,e^b=ei,e^c=ei,e^d=ej,g^c=gj,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250612, #72250889, #72257738
maximal normal subgroup of	#72250612, #72250889, #72257738

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.7);;
gap> SetPower(coll, 4, F.5*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);