#510044

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

549 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417240);