#504684

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411880);