#6435678

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1602 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)(11,15)(14,16)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,31,26,32);(10,13)(14,16)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(9,11,12,15)(10,14,13,16);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6342874);