#487270

order	512
nilpotency class	4
isoclinism	isoclinic class of #486883
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	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,13,18,43)(4,20,22,54)(5,26,29,12)(6,32,19,52)(8,34,9,15)(10,27,36,17)(11,38,40,16)(21,42,49,62)(23,57,56,39)(24,59,25,47)(28,35,58,63)(30,46,31,60)(33,37,51,45)(41,48,44,61)(50,64,55,53);(1,3,8,19)(2,10,34,45)(4,21,24,38)(5,27,30,33)(6,7,18,9)(11,39,62,63)(12,43,60,32)(13,46,52,26)(14,36,15,37)(16,22,49,25)(17,31,51,29)(20,53,59,41)(23,50,28,48)(35,40,57,42)(44,54,64,47)(55,58,61,56);(1,4,9,25)(2,11,15,42)(3,16,6,21)(5,28,31,56)(7,22,8,24)(10,35,37,39)(12,44,46,53)(13,47,32,20)(14,40,34,62)(17,50,33,61)(18,38,19,49)(23,29,58,30)(26,41,60,64)(27,55,51,48)(36,63,45,57)(43,59,52,54);(4,22)(11,40)(16,38)(20,54)(21,49)(23,56)(24,25)(28,58)(35,63)(39,57)(41,44)(42,62)(47,59)(48,61)(50,55)(53,64)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^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,e^b=eg,f^b=fhi,d^c=dg,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, 394466);