#7505957

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7413153);