#72282011

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 9633 conjugacy classes of subgroups; only the 3365 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)(5,6,10,9)(7,11,12,8)(13,14,18,17)(15,16,20,19)(21,22)(23,24)(25,27)(26,28);(1,2,4,3)(5,7,10,12)(6,11,9,8)(13,15,18,20)(14,16,17,19)(21,23)(22,26)(24,27)(25,28);(2,3)(5,8)(6,12)(7,9)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28);(1,3,4,2)(5,8)(6,12)(7,9)(10,11)(13,15)(14,19)(16,17)(18,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,24,25,26)(22,23,27,28);(5,9,10,6)(7,11,12,8)(13,17,18,14)(15,16,20,19)(21,23)(22,24)(25,28)(26,27)

Polycyclic presentation:

2^10|a^2=i,b^2=g,d^2=gi,e^2=ghi,f^2=i,b^a=bh,c^a=cgh,c^b=cg,d^a=dhi,d^b=di,d^c=dhij,e^b=eghj,f^b=fi,f^e=fgh

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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