#487201

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	#392	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 711 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,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, 394397);