#494027

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂³	C₂²×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401223);