#502739

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×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

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

normal subgroups only

This group has 870 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,37)(24,32)(25,33)(26,34)(27,35)(28,36)(29,39)(30,40)(31,41)(38,43)(42,44);(6,9)(10,11)(13,15,16,24)(14,19,20,32)(17,23,27,41)(18,30,28,26)(21,37,35,31)(22,40,36,34)(25,43,42,29)(33,38,44,39);(1,2,3,4)(14,20)(15,23)(17,29)(18,28)(19,31)(21,38)(24,41)(25,26)(27,43)(30,42)(32,37)(33,40)(34,44)(35,39);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=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, 409935);