#508169

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415365);