#512082

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419278);