#506325

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

474 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,16)(12,26)(13,20)(14,19)(15,21)(17,28)(18,33)(22,49)(23,37)(24,40)(25,39)(27,57)(29,58)(30,43)(31,44)(32,47)(34,42)(35,48)(36,51)(38,52)(41,56)(45,61)(46,62)(50,66)(53,67)(54,65)(55,70)(59,68)(60,69)(63,71)(64,72);(2,5)(6,7)(9,11,14,25,33,52,61,59,15,23,35,53,13,24,32,55)(10,16,19,39,18,38,45,68,21,37,48,67,20,40,47,70)(12,27,29,54,31,63,64,50,30,60,62,51,28,56,34,22)(17,41,42,49,26,57,58,65,44,71,72,66,43,69,46,36);(9,12,15,30)(10,17,21,44)(11,22,23,50)(13,31,33,28)(14,34,35,64)(16,36,37,65)(18,43,20,26)(19,46,48,58)(24,54,52,51)(25,56,53,63)(27,59,60,55)(29,61,62,32)(38,66,40,49)(39,69,67,57)(41,68,71,70)(42,45,72,47);(1,3,4,8)(2,6,5,7)(11,23)(16,37)(22,50)(24,52)(25,53)(27,60)(36,65)(38,40)(39,67)(41,71)(49,66)(51,54)(55,59)(56,63)(57,69)(68,70)

Polycyclic presentation:

2^9|a^2=e,b^2=g,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, 413521);