#506314

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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 ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 150 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)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,21)(17,22)(18,20)(23,30)(24,32)(25,33)(26,31)(27,36)(28,35)(29,34)(37,42)(38,41)(39,43)(40,44);(6,9)(10,11)(13,15,16,24,28,39,40,38,18,26,29,37,17,25,27,23)(14,19,21,32,35,43,44,41,20,31,34,42,22,33,36,30);(2,4)(15,23)(16,27)(17,28)(19,30)(21,36)(22,35)(24,25)(26,38)(29,40)(31,41)(32,33)(34,44)(37,39)(42,43);(5,7,8,12)(6,10,9,11)(14,20)(19,31)(21,34)(22,35)(30,41)(32,42)(33,43)(36,44)

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=e,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, 413510);