#483511

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₂×D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×C₈×D₈
prime radical	#3414	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

477 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,d^a=di,e^a=eg,g^a=gi,e^b=eg,g^b=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320299, #72320727, #72320780
maximal normal subgroup of	#72320299, #72320727, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390707);