#486083

order	512
nilpotency class	4
isoclinism	isoclinic class of #486082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18178
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 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)(5,23)(6,28)(7,12)(8,15)(10,27)(11,20)(14,42)(16,50)(17,43)(18,51)(19,34)(21,54)(22,56)(24,40)(25,41)(26,57)(29,37)(30,49)(31,48)(32,58)(33,53)(35,55)(36,44)(38,60)(39,47)(45,62)(46,61)(52,64)(59,63);(1,3,7,17)(2,8,12,31)(4,18,19,52)(5,24,25,58)(6,29,20,16)(9,33,34,63)(10,37,39,50)(11,40,28,32)(13,41,43,23)(14,44,45,22)(15,47,48,27)(21,49,26,46)(30,56,61,36)(35,42,38,62)(51,57,64,54)(53,60,59,55);(4,19)(5,25)(9,34)(14,45)(15,48)(18,52)(22,44)(24,58)(27,47)(30,61)(33,63)(35,38)(36,56)(42,62)(51,64)(54,57);(1,4,20,44,7,19,6,22)(2,9,11,36,12,34,28,56)(3,14,29,52,17,45,16,18)(5,26,47,55,25,21,27,60)(8,30,32,33,31,61,40,63)(10,38,23,57,39,35,41,54)(13,42,37,64,43,62,50,51)(15,49,58,53,48,46,24,59)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250797, #72307849
maximal normal subgroup of	#72250797, #72307849
maximal quotient of	#72257005

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393279);