#6368755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
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

1474 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)(3,11)(4,8)(5,19)(6,9)(7,14)(10,22)(12,20)(13,29)(15,27)(16,30)(17,28)(18,25)(21,26)(23,32)(24,31)(33,34)(35,45)(36,51)(37,42)(38,55)(39,43)(40,49)(41,54)(44,56)(46,60)(47,63)(48,58)(50,59)(52,57)(53,61)(62,64);(1,3,6,15)(2,7,9,12)(4,16,18,32)(5,20,22,14)(8,24,25,29)(10,27,19,11)(13,21,31,17)(23,28,30,26)(33,35)(34,40)(36,52)(37,46)(38,50)(39,48)(41,60)(42,62)(43,59)(44,58)(45,55)(47,54)(49,56)(51,63)(53,64)(57,61);(1,2)(3,12)(4,8)(5,10)(6,9)(7,15)(11,20)(13,30)(14,27)(16,29)(17,26)(18,25)(19,22)(21,28)(23,31)(24,32)(33,36,39,37)(34,41,43,61)(35,46,48,52)(38,53,56,54)(40,57,59,60)(42,55,51,44)(45,62,58,63)(47,49,64,50);(1,4,6,18)(2,8,9,25)(3,13,15,31)(5,21,22,17)(7,23,12,30)(10,28,19,26)(11,29,27,24)(14,32,20,16)(33,37,39,36)(34,42,43,51)(35,47,48,64)(38,53,56,54)(40,57,59,60)(41,55,61,44)(45,63,58,62)(46,49,52,50);(2,9)(7,12)(8,25)(10,19)(11,27)(23,30)(24,29)(26,28)(34,43)(35,48)(36,37)(44,55)(49,50)(53,54)(57,60)(62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306633, #72306654, #72306668
maximal normal subgroup of	#72306633, #72306654, #72306668

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275951);