#72334942

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

247129 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|f^2=i,c^b=ci,d^a=di,d^c=di,f^b=fj,f^c=fg,f^d=fh,f^e=fg

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334943, #72337233
maximal normal subgroup of	#72334943, #72337233
maximal quotient of	#72334982, #72335079, #72337244

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