#72335970

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 6877 conjugacy classes of subgroups; only the 3541 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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=ghi,e^2=ghj,f^2=gj,b^a=bg,c^a=cij,d^a=dgh,d^c=dij,e^a=eghij,e^c=eij,e^d=eij,f^a=fg,f^d=fgij,f^e=fi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336001
maximal normal subgroup of	#72336001
maximal quotient of	#72336469

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