#487210

order	512
nilpotency class	4
isoclinism	isoclinic class of #486819
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

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

normal subgroups only

This group has 710 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)(3,15,20,38)(4,10,23,40)(5,11,26,43)(6,27,31,12)(8,14,16,32)(9,28,36,19)(17,44,49,62)(18,46,24,41)(21,47,48,64)(22,54,56,39)(25,37,55,42)(29,59,52,34)(30,57,50,45)(33,60,63,51)(35,58,61,53);(1,3,16,48)(2,9,32,61)(4,5,18,49)(6,28,50,53)(7,20,8,21)(10,37,46,63)(11,34,62,54)(12,38,57,47)(13,36,14,35)(15,45,64,27)(17,23,26,24)(19,30,58,31)(22,55,52,51)(25,29,60,56)(33,40,42,41)(39,43,59,44);(1,4,8,24)(2,10,14,41)(3,17,21,5)(6,29,30,22)(7,23,16,18)(9,33,35,37)(11,15,44,47)(12,39,45,34)(13,40,32,46)(19,25,53,51)(20,49,48,26)(27,54,57,59)(28,55,58,60)(31,52,50,56)(36,63,61,42)(38,62,64,43);(1,5,8,17)(2,11,14,44)(3,18,21,23)(4,20,24,48)(6,25,30,51)(7,26,16,49)(9,34,35,39)(10,38,41,64)(12,42,45,63)(13,43,32,62)(15,46,47,40)(19,52,53,56)(22,28,29,58)(27,37,57,33)(31,55,50,60)(36,59,61,54)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,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, 394406);