#487248

order	512
nilpotency class	4
isoclinism	isoclinic class of #486859
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

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

normal subgroups only

This group has 689 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:

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

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,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, 394444);