#504776

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
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 ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

571 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi,e^d=eh,f^d=fh,h^d=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, 411972);