#510035

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

504 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417231);