#498460

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405656);