#499171

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,4)(9,10,17,24)(11,25,31,55)(12,32,34,53)(13,22,40,21)(14,41,15,20)(16,43,39,23)(18,42,49,27)(19,37,51,36)(26,58,59,45)(28,48,29,56)(30,46,44,47)(33,64,61,57)(35,63,67,54)(38,69,68,52)(50,60,70,66)(62,71,65,72);(5,6)(7,8)(9,11,17,31)(10,18,24,49)(12,33,34,61)(13,29,40,28)(14,42,15,27)(16,44,39,30)(19,50,51,70)(20,47,41,46)(21,55,22,25)(23,56,43,48)(26,36,59,37)(32,58,53,45)(35,62,67,65)(38,60,68,66)(52,71,69,72)(54,57,63,64);(2,4)(6,8)(9,12,13,35,39,68,14,36,17,34,40,67,16,38,15,37)(10,19,20,52,43,63,21,32,24,51,41,69,23,54,22,53)(11,26,27,60,44,65,28,61,31,59,42,66,30,62,29,33)(18,45,25,57,56,72,46,70,49,58,55,64,48,71,47,50);(12,34)(19,51)(26,59)(32,53)(33,61)(35,67)(36,37)(38,68)(45,58)(50,70)(52,69)(54,63)(57,64)(60,66)(62,65)(71,72)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406367);