#508013

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1060 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)(13,14,17,23)(15,25,27,20)(16,22,19,33)(18,21,31,39)(24,30,38,32)(26,37,29,35)(28,40,43,34)(36,42,41,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,29)(17,27)(18,28)(19,26)(21,34)(22,37)(23,25)(24,36)(30,44)(31,43)(32,42)(33,35)(38,41)(39,40);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(16,19)(18,32)(20,34)(22,24)(23,39)(25,40)(26,29)(28,42)(30,31)(33,38)(35,41)(36,37)(43,44);(7,11)(10,12)(13,16,17,19)(14,22,23,33)(15,26,27,29)(18,30,31,32)(20,35,25,37)(21,38,39,24)(28,42,43,44)(34,36,40,41)

Polycyclic presentation:

2^9|a^2=i,d^2=i,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, 415209);