#6633353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	#37301	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 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)(10,14)(11,12)(17,18)(19,26)(20,22)(21,25)(23,30)(24,29)(27,32)(28,31);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(18,20)(19,27)(25,30)(29,31);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,31)(25,32)(26,30);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,20,22,18)(19,28,27,24)(21,25,23,30)(26,29,32,31);(2,6)(5,8)(17,21,22,23)(18,25,20,30)(19,29,27,31)(24,32,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540549);