#506254

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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

628 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 158 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)(7,11)(8,9)(10,12)(14,19)(20,32)(21,33)(22,34)(23,39)(35,38)(36,44)(37,40);(6,9)(10,11)(13,14,17,21,28,34,42,40,16,19,27,33,18,22,30,37)(15,23,25,38,31,44,43,32,24,39,41,35,26,36,29,20);(2,4)(13,15,16,24)(14,20,19,32)(17,29,27,43)(18,31,28,26)(21,36,33,44)(22,38,34,35)(23,40,39,37)(25,42,41,30);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=ei,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gh,h^c=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, 413450);