#506434

order	512
nilpotency class	4
isoclinism	isoclinic class of #500848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

482 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=i,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,d^b=di,f^c=fi,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, 413630);