#487296

order	512
nilpotency class	4
isoclinism	isoclinic class of #486875
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×C₄⋊C₄	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 687 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,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, 394492);