#487194

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	#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

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

normal subgroups only

This group has 706 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)(3,15,20,37)(4,10,23,39)(5,11,26,42)(6,27,31,12)(8,14,16,32)(9,28,36,19)(17,43,48,62)(18,45,24,40)(21,46,47,64)(22,53,55,38)(25,56,57,41)(29,60,51,34)(30,58,49,44)(33,54,63,50)(35,59,61,52);(1,3,16,47)(2,9,32,61)(4,17,18,26)(5,23,48,24)(6,28,49,52)(7,20,8,21)(10,33,45,41)(11,38,62,60)(12,37,58,46)(13,36,14,35)(15,44,64,27)(19,30,59,31)(22,54,51,25)(29,57,55,50)(34,42,53,43)(39,63,40,56);(1,4,8,24)(2,10,14,40)(3,17,21,5)(6,29,30,22)(7,23,16,18)(9,33,35,56)(11,15,43,46)(12,38,44,34)(13,39,32,45)(19,25,52,50)(20,48,47,26)(27,53,58,60)(28,57,59,54)(31,51,49,55)(36,63,61,41)(37,62,64,42);(1,5,8,17)(2,11,14,43)(3,18,21,23)(4,20,24,47)(6,25,30,50)(7,26,16,48)(9,34,35,38)(10,37,40,64)(12,41,44,63)(13,42,32,62)(15,45,46,39)(19,51,52,55)(22,28,29,59)(27,56,58,33)(31,57,49,54)(36,60,61,53)

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,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, 394390);