#6349134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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

2586 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)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(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);(1,3,7,16)(2,8,12,22)(4,14,17,28)(5,15,6,20)(9,21,23,31)(10,13,11,26)(18,29,19,30)(24,27,25,32)(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)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,25)(12,23)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32)(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);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(34,42)(36,51)(40,55)(43,44)(45,56)(46,61)(52,53)(62,63);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,17,19)(8,13,22,26)(9,24,23,25)(14,29,28,30)(21,27,31,32)(34,42)(40,55)(41,57)(43,44)(45,56)(50,58)(54,60)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306547, #72306699, #72306740
maximal normal subgroup of	#72306547, #72306699, #72306740

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256330);