#6637402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636852
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	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,18)(20,25,28,23)(22,26,30,24)(27,29,32,31);(11,15)(13,16)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,19)(20,28)(24,26)(29,31);(2,5)(6,7)(10,14)(13,16)(18,19)(23,25)(24,26)(27,32)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6544598);