#7623936

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ #2960 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1230 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531132);