#7504659

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2068 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,23,28,32)(20,30,22,24)(26,29,31,27);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,21,28)(18,23,25,32)(20,29,22,27)(24,26,30,31);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(17,18)(19,26)(20,24)(21,25)(22,30)(23,27)(28,31)(29,32);(2,3)(6,7)(10,14)(13,16)(17,18)(19,23)(20,30)(21,25)(22,24)(26,27)(28,32)(29,31);(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7411855);