#513304

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=ei,d^2=ei,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420500);