#508953

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 568 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,24,29,36,43,49,47,26,28,38,42,25,30,37,45)(23,31,32,46,39,52,51,40,34,48,50,41,33,44,35,27);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19);(2,4)(8,16)(11,18)(14,20)(17,19)(21,23,26,34)(22,27,28,40)(24,35,38,51)(25,39,36,33)(29,44,42,52)(30,46,43,41)(31,47,48,45)(32,49,50,37);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(27,40)(29,42)(30,43)(31,48)(41,46)(44,52)(45,47)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=i,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, 416149);