#72234811

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 10003 conjugacy classes of subgroups; only the 3353 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:

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

Polycyclic presentation:

2^10|b^2=hj,e^2=hj,f^2=i,b^a=bh,d^b=dj,e^a=eg,e^b=eghi,e^c=eghi,e^d=eij,f^a=fhj,f^b=fh,f^c=fij,f^e=fij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234830, #72332285
maximal normal subgroup of	#72234830, #72332285
maximal quotient of	#72333262

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