#508925

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
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	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416121);