#72328880

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

At least 10218 conjugacy classes of subgroups; only the 3313 normal subgroups are shown.

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

Polycyclic presentation:

2^10|c^2=hij,f^2=hj,b^a=bj,c^a=ci,c^b=ch,d^a=di,d^b=dhij,d^c=dhij,e^c=eij,e^d=eij,f^d=fhj,f^e=fg

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329012
maximal normal subgroup of	#72329012
maximal quotient of	#72331213

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