#495720

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#447
prime radical	#33538	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1009 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225388, #72225466, #72225467
maximal normal subgroup of	#72225388, #72225466, #72225467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402916);