#6793216

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6700412);