#72335513

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

33623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=h,f^2=ghi,b^a=bg,c^a=cj,c^b=chi,d^b=dghj,d^c=dghj,e^b=egj,e^c=egj,f^a=fj,f^c=fh,f^d=fhij,f^e=fgj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335547
maximal normal subgroup of	#72335547
maximal quotient of	#72336065

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