#501113

order	512
nilpotency class	4
isoclinism	isoclinic class of #501108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24404
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1070 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)(3,15)(4,10)(5,24)(6,12,29,42)(7,13)(8,14)(9,18,34,26)(11,25,41,28)(16,46)(17,33)(19,43)(20,53)(21,39)(22,38,27,58)(23,40)(30,59)(31,45)(32,48,61,51)(35,50,44,52)(36,54,62,56)(37,55,63,57)(47,60)(49,64);(1,3,8,19)(2,9,14,34)(4,20,23,47)(5,17,7,30)(6,26,29,18)(10,36,40,62)(11,33,41,59)(12,43,42,15)(13,44,24,35)(16,31,49,21)(22,56,27,54)(25,52,28,50)(32,45,61,39)(37,46,63,64)(38,60,58,53)(48,55,51,57);(1,4)(2,10)(3,16)(5,21)(6,27)(7,31)(8,23)(9,32)(11,37)(12,38)(13,45)(14,40)(15,46)(17,47)(18,51)(19,49)(20,30)(22,29)(24,39)(25,57)(26,48)(28,55)(33,60)(34,61)(35,62)(36,44)(41,63)(42,58)(43,64)(50,54)(52,56)(53,59);(1,5,8,7)(2,11,14,41)(3,17,19,30)(4,21,23,31)(6,28,29,25)(9,33,34,59)(10,37,40,63)(12,13,42,24)(15,35,43,44)(16,47,49,20)(18,52,26,50)(22,57,27,55)(32,60,61,53)(36,46,62,64)(38,45,58,39)(48,54,51,56)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312282, #72312293, #72312372, …
maximal normal subgroup of	#72312282, #72312293, #72312372, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408309);