#494973

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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 ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

586 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,26,16,21)(17,23,20,36)(19,33,32,37)(22,28,41,30)(25,31,43,35)(27,40,29,38)(34,39,44,42);(1,3,4,2)(6,9)(10,12)(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);(5,7,8,11)(6,10,9,12)(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=g,c^2=i,d^2=e,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, 402169);