#6356937

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1447 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264133);