#6355482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60961	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1507 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)(6,20)(7,11)(8,14)(13,27)(15,23)(16,22)(17,24)(18,25)(19,26)(21,29)(28,31)(30,32)(33,34,39,44)(35,40,46,54)(36,49,51,58)(37,43,38,42)(41,52,57,50)(45,59,60,64)(47,56,48,55)(53,62,63,61);(1,3,6,14,7,16,5,15)(2,8,10,12,11,23,20,22)(4,13,17,29,19,30,18,28)(9,21,25,27,26,31,24,32)(33,35)(34,40)(36,45)(37,47)(38,48)(39,46)(41,53)(42,55)(43,56)(44,54)(49,59)(50,61)(51,60)(52,62)(57,63)(58,64);(1,4)(2,9)(3,13)(5,18)(6,17)(7,19)(8,21)(10,25)(11,26)(12,27)(14,29)(15,28)(16,30)(20,24)(22,32)(23,31)(33,36)(34,41)(35,45)(37,50)(38,52)(39,51)(40,53)(42,49)(43,58)(44,57)(46,60)(47,61)(48,62)(54,63)(55,59)(56,64);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,19,18)(8,22,23,12)(9,24,26,25)(13,28,30,29)(21,27,31,32)(35,46)(40,54)(45,60)(47,48)(53,63)(55,56)(59,64)(61,62);(1,5,7,6)(2,10,11,20)(3,15,16,14)(4,18,19,17)(8,12,23,22)(9,25,26,24)(13,28,30,29)(21,27,31,32)(33,37,39,38)(34,42,44,43)(35,47,46,48)(36,50,51,52)(40,55,54,56)(41,49,57,58)(45,61,60,62)(53,59,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306994, #72306999, #72307005, …
maximal normal subgroup of	#72306994, #72306999, #72307005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262678);