#495988

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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 ◁ #376 ◁ #1726 ◁ 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403184);