#6350970

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	#537	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2527 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^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, 6258166);