#7616280

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=fg,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 7523476);