#7607058

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 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)(9,10,13,12)(11,14,16,15)(18,23)(20,28)(21,29)(24,31)(25,27)(30,32);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,18,21,23,22,24,29,31)(19,20,27,28,26,30,25,32);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,19)(18,20)(21,27)(22,26)(23,28)(24,30)(25,29)(31,32);(2,6)(5,7)(11,16)(14,15)(18,24)(19,26)(23,31)(25,27);(2,6)(5,7)(17,20,29,32,22,30,21,28)(18,25,31,26,24,27,23,19)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=hi,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,e^b=ei,d^c=di

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, 7514254);