#6775766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682962);