#506422

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413618);