#6433564

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340760);