#6364574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6364250
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306850, #72306856, #72306858
maximal normal subgroup of	#72306850, #72306856, #72306858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271770);