#499428

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,12)(8,10)(9,11)(13,14,17,21,29,36,43,44,18,22,31,38,16,20,28,35)(15,23,25,32,24,39,41,34,27,40,42,33,26,37,30,19);(1,2)(3,4)(5,7,9,8)(6,10,11,12);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,24)(17,30)(18,27)(20,32)(21,37)(22,34)(23,35)(25,28)(26,29)(31,41)(33,36)(38,39)(40,44)(42,43);(13,16,18,29)(14,20,22,36)(15,24,27,26)(17,28,31,43)(19,32,34,33)(21,35,38,44)(23,39,40,37)(25,41,42,30)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406624);