#502946

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 717 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,c^2=f,d^2=e,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,d^c=di,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410142);