#497924

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

Polycyclic presentation:

2^9|a^2=g,c^2=efhi,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, 405120);