#487221

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 ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 736 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)(3,13,17,35)(4,9,19,32)(5,22,26,10)(7,12,27,37)(8,23,30,16)(14,38,43,63)(15,39,45,64)(18,47,41,61)(20,49,48,33)(21,34,40,57)(24,53,55,62)(25,51,56,36)(28,54,59,42)(29,52,60,44)(31,50,58,46);(1,3,6,17)(2,8,11,30)(4,18,19,41)(5,23,26,16)(7,15,27,45)(9,31,32,58)(10,35,22,13)(12,29,37,60)(14,40,43,21)(20,50,48,46)(24,54,55,42)(25,52,56,44)(28,57,59,34)(33,61,49,47)(36,64,51,39)(38,62,63,53);(1,4,7,21)(2,9,12,34)(3,14,15,18)(5,24,25,20)(6,19,27,40)(8,28,29,31)(10,33,36,62)(11,32,37,57)(13,38,39,47)(16,46,44,42)(17,43,45,41)(22,49,51,53)(23,50,52,54)(26,55,56,48)(30,59,60,58)(35,63,64,61);(3,15)(4,19)(8,29)(9,32)(13,39)(14,41)(16,44)(17,45)(18,43)(20,48)(21,40)(23,52)(24,55)(28,58)(30,60)(31,59)(33,49)(34,57)(35,64)(38,61)(42,50)(46,54)(47,63)(53,62)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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=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, 394417);