#494208

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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)(15,23)(16,25)(17,28)(19,30)(21,31)(22,34)(24,35)(26,42)(27,38)(29,41)(32,44)(33,39)(36,37)(40,43);(6,9)(10,12)(13,14)(15,24)(16,21)(17,22)(18,20)(19,27)(23,35)(25,31)(26,40)(28,34)(29,33)(30,38)(32,36)(37,44)(39,41)(42,43);(13,15,25,37,17,26,29,38,18,27,41,42,28,36,16,23)(14,19,31,43,22,32,33,35,20,24,39,44,34,40,21,30);(5,7,8,11)(6,10,9,12)(14,20)(19,24)(21,33)(22,34)(30,35)(31,39)(32,40)(43,44)

Polycyclic presentation:

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

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, 401404);