#494978

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,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=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, 402174);