#487230

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
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 ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 737 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)(3,13,18,38)(4,9,22,36)(5,24,28,10)(7,12,14,29)(8,25,32,17)(15,40,44,63)(16,41,42,64)(19,48,49,61)(20,34,23,37)(21,51,53,35)(26,56,50,62)(27,54,43,39)(30,57,59,45)(31,55,58,46)(33,52,60,47);(1,3,14,42)(2,8,29,58)(4,19,20,44)(5,25,43,46)(6,18,7,16)(9,33,34,59)(10,38,54,41)(11,32,12,31)(13,39,64,24)(15,22,49,23)(17,27,55,28)(21,52,50,45)(26,57,53,47)(30,36,60,37)(35,61,56,40)(48,62,63,51);(1,4,7,23)(2,9,12,37)(3,15,16,19)(5,26,27,21)(6,22,14,20)(8,30,31,33)(10,35,39,62)(11,36,29,34)(13,40,41,48)(17,47,46,45)(18,44,42,49)(24,51,54,56)(25,52,55,57)(28,50,43,53)(32,59,58,60)(38,63,64,61);(3,16)(4,20)(8,31)(9,34)(13,41)(15,44)(17,46)(18,42)(19,49)(21,50)(22,23)(25,55)(26,53)(30,59)(32,58)(33,60)(35,56)(36,37)(38,64)(40,63)(45,57)(47,52)(48,61)(51,62)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^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=dgi,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, 394426);