#497922

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

624 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)(6,9)(10,12)(13,14)(15,24)(16,20)(17,30)(18,23)(19,28)(21,26)(22,29)(25,37)(27,38)(31,35)(32,42)(33,43)(34,44)(36,40)(39,41);(5,6)(7,10)(8,9)(11,12)(15,25)(19,32)(24,37)(27,41)(28,42)(33,34)(38,39)(43,44);(5,7,8,11)(6,10,9,12)(13,15,26,43,16,27,31,42,18,25,40,44,29,41,17,28)(14,19,30,39,22,34,36,37,23,32,35,38,20,33,21,24);(6,9)(10,12)(13,16,18,29)(14,20,23,22)(15,27,25,41)(17,26,31,40)(19,33,32,34)(21,35,36,30)(24,38,37,39)(28,43,42,44)

Polycyclic presentation:

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

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, 405118);