#6368761

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

1483 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,e^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	#72306666, #72306696, #72306703
maximal normal subgroup of	#72306666, #72306696, #72306703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275957);