#6793195

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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)(17,18,21,25)(19,26,22,23)(20,28,30,31)(24,29,32,27);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,31,32,28);(10,13)(14,15)(17,20,21,30)(18,24,25,32)(19,27,22,29)(23,28,26,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,14,13,15)(18,25)(23,26)(24,32)(28,31);(1,4,5,3)(2,6,7,8)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6700391);