#502756

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 810 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,22)(12,27)(13,19)(14,33)(15,21)(16,20)(17,24)(18,30)(23,54)(25,41)(26,53)(28,62)(29,47)(31,61)(32,48)(34,68)(35,69)(36,46)(37,49)(38,43)(39,45)(40,57)(42,58)(44,59)(50,66)(51,65)(52,56)(55,70)(60,63)(64,71)(67,72);(2,5)(6,7)(9,11)(10,17)(12,28)(13,24)(14,25)(15,35)(16,38)(18,44)(19,22)(20,41)(21,50)(23,31)(26,37)(27,60)(29,63)(30,64)(32,66)(33,43)(34,58)(36,56)(39,59)(40,46)(42,49)(45,71)(47,62)(48,69)(51,54)(52,72)(53,68)(55,65)(57,67)(61,70);(9,12,14,30,13,29,16,39)(10,18,20,27,19,45,33,47)(11,23,25,56,24,55,38,57)(15,36,34,65,32,67,37,31)(17,40,41,70,22,52,43,54)(21,51,49,46,48,61,68,72)(26,59,35,28,58,64,66,63)(42,60,69,71,53,62,50,44);(1,3,4,8)(2,6,5,7)(10,19)(17,22)(18,45)(20,33)(21,48)(27,47)(40,52)(41,43)(42,53)(44,71)(46,72)(49,68)(50,69)(51,61)(54,70)(60,62)

Polycyclic presentation:

2^9|c^2=f,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gh,h^b=hi,g^c=gh,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, 409952);