#7508485

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,28,31)(20,29,21,25)(26,30,32,27);(1,3)(2,6)(4,7)(5,8)(17,19,22,28)(18,24,23,31)(20,30,21,27)(25,26,29,32);(9,11,12,16)(10,13,14,15)(17,20,22,21)(18,25,23,29)(19,26,28,32)(24,27,31,30);(2,7)(3,8)(10,14)(13,15)(17,21,22,20)(18,25,23,29)(19,27,28,30)(24,26,31,32);(1,4,5,6)(2,3,7,8)(17,18,22,23)(19,24,28,31)(20,25,21,29)(26,27,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,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, 7415681);