#492529

order	512
nilpotency class	4
isoclinism	isoclinic class of #492528
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₈²
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=de,e^b=eg,g^b=gi,f^c=fh,h^c=hi,e^d=eg,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399725);