#487223

order	512
nilpotency class	4
isoclinism	isoclinic class of #486803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394419);