#506456

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

557 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)(13,14,16,20)(15,23,25,42)(17,30,28,41)(18,33,29,22)(19,35,24,26)(21,38,37,32)(27,43,34,39)(31,40,44,36);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,24)(16,20)(17,21)(18,22)(19,25)(23,35)(26,42)(27,40)(28,37)(29,33)(30,32)(31,43)(34,36)(38,41)(39,44);(2,4)(13,15,16,25)(14,19,20,24)(17,31,28,44)(18,34,29,27)(21,39,37,43)(22,40,33,36)(23,41,42,30)(26,38,35,32);(5,8,9,7)(6,11,12,10)(14,20)(19,24)(21,37)(22,33)(23,42)(30,41)(36,40)(39,43)

Polycyclic presentation:

2^9|a^2=e,b^2=eg,c^2=i,d^2=ei,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, 413652);