#6697353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18,23,24)(19,27,31,25)(20,22,26,21)(28,30,29,32);(10,13)(11,15)(17,19)(18,25)(20,30)(21,29)(22,28)(23,31)(24,27)(26,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,28)(21,24)(23,26)(25,32)(27,30)(29,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21)(18,26)(19,29)(20,24)(22,23)(25,32)(27,30)(28,31);(17,22,23,21)(18,26,24,20)(19,30,31,32)(25,28,27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6604549);