#7607428

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ghi,d^2=i,e^2=fgi,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^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, 7514624);