#7512002

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	#30458	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7419198);