#6351052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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	#62952	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258248);