#487246

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	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

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

normal subgroups only

This group has 689 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)(4,20,23,55)(5,25,28,12)(6,31,19,52)(8,34,9,15)(10,26,36,17)(11,38,41,16)(21,53,24,47)(22,56,59,54)(27,35,60,63)(29,46,30,61)(32,42,48,39)(33,37,51,45)(40,49,44,57)(50,62,58,64);(1,3,8,19)(2,10,34,45)(4,16,21,48)(5,26,29,33)(6,7,18,9)(11,35,39,56)(12,43,61,31)(13,46,52,25)(14,36,15,37)(17,30,51,28)(20,44,53,62)(22,57,27,58)(23,38,24,32)(40,47,64,55)(41,63,42,54)(49,60,50,59);(1,4,9,24)(2,11,15,42)(3,16,6,32)(5,27,30,59)(7,23,8,21)(10,35,37,54)(12,44,46,64)(13,47,31,20)(14,41,34,39)(17,50,33,57)(18,38,19,48)(22,28,60,29)(25,40,61,62)(26,58,51,49)(36,63,45,56)(43,53,52,55);(4,21)(11,39)(16,48)(20,53)(22,27)(23,24)(32,38)(35,56)(40,64)(41,42)(44,62)(47,55)(49,50)(54,63)(57,58)(59,60)

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,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, 394442);