#6355106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355044
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1485 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262302);