#6456174

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1681 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,27,28)(21,24,31,29)(25,30,32,26);(2,6)(7,8)(17,19,22,27)(18,23,20,28)(21,30,31,26)(24,32,29,25);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,18)(19,25,27,32)(21,24,31,29)(23,30,28,26);(1,4,5,3)(2,8,6,7)(19,26)(21,31)(23,25)(24,29)(27,30)(28,32);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6363370);