#72327353

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 9190 conjugacy classes of subgroups; only the 3357 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:

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

Polycyclic presentation:

2^10|c^2=gh,d^2=ghi,f^2=ij,b^a=bg,c^b=cgi,d^c=di,e^c=eghj,e^d=eghj,f^a=fgi,f^c=fgj,f^e=fgi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327472
maximal normal subgroup of	#72327472

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