#506404

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

679 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,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, 413600);