#511858

order	512
nilpotency class	4
isoclinism	isoclinic class of #511856
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2043 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)(3,13)(4,9)(5,22)(6,11)(7,12)(8,16)(10,24)(14,39)(15,42)(17,38)(18,36)(19,48)(20,31)(21,32)(23,53)(25,35)(26,41)(27,56)(28,45)(29,47)(30,50)(33,54)(34,51)(37,44)(40,60)(43,52)(46,61)(49,59)(55,63)(57,62)(58,64);(1,3,7,17)(2,8,12,28)(4,18,21,14)(5,15,24,44)(6,16,25,45)(9,29,32,26)(10,27,22,52)(11,13,35,38)(19,46,50,40)(20,47,51,41)(23,43,54,56)(30,57,48,55)(31,36,34,39)(33,37,53,42)(49,63,64,62)(58,60,59,61);(1,4,7,21)(2,9,12,32)(3,14,17,18)(5,19,24,50)(6,20,25,51)(8,26,28,29)(10,30,22,48)(11,34,35,31)(13,36,38,39)(15,40,44,46)(16,41,45,47)(23,49,54,64)(27,55,52,57)(33,59,53,58)(37,60,42,61)(43,62,56,63);(1,5)(2,10,12,22)(3,15)(4,19)(6,23)(7,24)(8,27,28,52)(9,30,32,48)(11,33,35,53)(13,37,38,42)(14,40)(16,43)(17,44)(18,46)(20,49)(21,50)(25,54)(26,55,29,57)(31,58,34,59)(36,60,39,61)(41,62)(45,56)(47,63)(51,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303825
maximal normal subgroup of	#72303825
maximal quotient of	#72303830, #72303838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419054);