#508415

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

533 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=i,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415611);