#7500968

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,3,4)(9,10)(11,19)(12,23)(13,15)(14,27)(16,22)(17,29)(18,26)(20,36)(21,34)(24,33)(25,31)(28,30)(32,37)(35,39)(38,40);(1,2)(3,4)(5,6)(7,8)(10,15)(16,30)(17,31)(18,27)(19,34)(23,33)(32,36)(35,40);(6,8)(9,11)(10,16)(12,20)(13,21)(14,28)(15,30)(17,32)(18,34)(19,27)(22,26)(23,40)(24,37)(25,39)(29,38)(31,36)(33,35);(2,4)(9,12,13,24)(10,17,15,31)(11,20,21,37)(14,29,26,25)(16,32,30,36)(18,23,27,33)(19,35,34,40)(22,39,28,38);(11,21)(16,30)(19,34)(20,37)(22,28)(32,36)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408164);