#6356656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
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

1458 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)(3,11,13,24)(4,8,16,22)(5,18,12,9)(7,19,20,15)(14,25,27,32)(17,29,26,23)(21,30,31,28)(33,34,39,44)(35,45,49,56)(36,50,53,59)(37,42,38,43)(40,48,57,47)(41,54,60,52)(46,58,51,55)(61,63,64,62);(1,3,12,19,6,13,5,15)(2,7,18,24,10,20,9,11)(4,14,26,30,16,27,17,28)(8,21,29,32,22,31,23,25)(33,35,38,48,39,49,37,47)(34,40,42,56,44,57,43,45)(36,51,54,62,53,46,52,63)(41,58,59,61,60,55,50,64);(3,13)(7,20)(11,24)(14,27)(15,19)(21,31)(25,32)(28,30)(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)(5,17)(6,16)(7,21)(9,23)(10,22)(11,25)(12,26)(13,27)(15,28)(18,29)(19,30)(20,31)(24,32)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,53,54)(40,56,57,45)(41,59,60,50)(46,62,51,63)(55,64,58,61);(4,16)(8,22)(14,27)(17,26)(21,31)(23,29)(25,32)(28,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,52,53,54)(40,56,57,45)(41,59,60,50)(46,62,51,63)(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,g^c=gi,e^d=ehi

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, 6263852);