#506449

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 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)(13,14,16,19)(15,23,24,41)(17,29,27,42)(18,32,28,22)(20,36,34,25)(21,38,35,31)(26,43,33,39)(30,40,44,37);(1,2,3,4)(5,7,9,8)(6,10,12,11)(14,19)(20,34)(21,35)(22,32)(23,41)(29,42)(37,40)(39,43);(2,4)(13,15,16,24)(14,20,19,34)(17,30,27,44)(18,33,28,26)(21,39,35,43)(22,40,32,37)(23,42,41,29)(25,38,36,31);(5,8,9,7)(6,11,12,10)(14,19)(20,34)(21,35)(22,32)(23,41)(29,42)(37,40)(39,43)

Polycyclic presentation:

2^9|a^2=e,b^2=egi,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,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413645);