#508935

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 170 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)(13,14,16,22,28,37,42,35,17,21,29,36,18,23,27,20)(15,19,25,32,39,44,40,30,24,31,38,43,26,33,41,34);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,34)(21,31)(22,32)(23,33)(27,41)(28,39)(29,38)(30,35)(36,43)(37,44)(40,42);(2,4)(7,11)(10,12)(14,20)(15,24)(16,27)(18,28)(19,30)(21,35)(22,23)(25,40)(29,42)(31,34)(32,44)(33,43)(36,37)(38,41);(5,7,8,11)(6,10,9,12)(14,21)(19,31)(20,35)(22,36)(23,37)(30,34)(32,43)(33,44)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,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, 416131);