#512453

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 218 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)(7,10)(8,9)(11,12)(13,14)(15,27)(16,21)(17,35)(18,38)(19,39)(20,26)(22,29)(23,28)(24,37)(25,36)(30,42)(31,41)(32,40)(33,44)(34,43);(6,9)(10,12)(13,15,28,34,19,31,37,32,20,29,18,30,36,33,17,16)(14,21,35,44,25,42,38,22,26,40,24,41,39,43,23,27);(1,2,3,4)(15,29)(16,32)(21,40)(22,27)(30,34)(31,33)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16,17,33,36,30,18,29,20,32,37,31,19,34,28,15)(14,22,23,42,39,44,24,21,26,27,38,43,25,41,35,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419649);