#6368952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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

1469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=h,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,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306654, #72306657, #72306693
maximal normal subgroup of	#72306654, #72306657, #72306693

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276148);