#7590413

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

1411 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,23)(20,29)(21,25)(24,30)(27,32)(28,31);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,23,22,20,25,29)(19,24,28,30,26,27,31,32);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(2,6)(5,7)(11,15)(13,16)(18,20)(19,26)(23,29)(28,31);(17,20,21,29,22,18,25,23)(19,27,28,32,26,24,31,30)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7497609);