#508459

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

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

489 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ei,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, 415655);