#503068

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410264);