#506453

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

561 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413649);