#6369143

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369070
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

1516 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306668, #72306670, #72306693
maximal normal subgroup of	#72306668, #72306670, #72306693

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276339);