#6349798

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349761
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62752	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2594 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)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,27)(28,31)(30,32)(33,34,39,42)(35,45,49,56)(36,50,51,58)(37,44,38,43)(40,48,55,47)(41,52,57,53)(46,60,61,54)(59,62,64,63);(1,3,7,15)(2,8,11,23)(4,13,17,28)(5,14,6,20)(9,21,25,32)(10,22,19,12)(16,27,18,30)(24,31,29,26)(33,35,39,49)(34,40,42,55)(36,46,51,61)(37,48,38,47)(41,54,57,60)(43,56,44,45)(50,64,58,59)(52,63,53,62);(1,4)(2,9)(3,13)(5,16)(6,18)(7,17)(8,21)(10,24)(11,25)(12,26)(14,27)(15,28)(19,29)(20,30)(22,31)(23,32)(33,36)(34,41)(35,46)(37,53)(38,52)(39,51)(40,54)(42,57)(43,50)(44,58)(45,59)(47,63)(48,62)(49,61)(55,60)(56,64);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,16,17,18)(8,22,23,12)(9,24,25,29)(13,27,28,30)(21,31,32,26)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64);(4,17)(9,25)(13,28)(16,18)(21,32)(24,29)(26,31)(27,30)(36,51)(41,57)(46,61)(50,58)(52,53)(54,60)(59,64)(62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306673, #72306681, #72306723
maximal normal subgroup of	#72306673, #72306681, #72306723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256994);