#6356580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
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

1469 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,15)(13,26)(14,22)(16,27)(17,23)(18,25)(19,24)(21,28)(29,31)(30,32)(33,34,39,44)(35,45,49,55)(36,50,52,57)(37,42,38,43)(40,48,56,47)(41,53,58,51)(46,60,62,54)(59,63,64,61);(1,3,5,14,7,16,6,15)(2,8,20,27,11,22,10,12)(4,13,19,30,18,29,17,28)(9,21,23,31,25,32,24,26)(33,35,38,48,39,49,37,47)(34,40,42,55,44,56,43,45)(36,46,53,63,52,62,51,61)(41,54,57,64,58,60,50,59);(33,36,39,52)(34,41,44,58)(35,46,49,62)(37,51,38,53)(40,54,56,60)(42,57,43,50)(45,59,55,64)(47,61,48,63);(1,4,7,18)(2,9,11,25)(3,13,16,29)(5,19,6,17)(8,21,22,32)(10,24,20,23)(12,26,27,31)(14,30,15,28)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,55,56,45)(41,57,58,50)(46,61,62,63)(54,64,60,59);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,17,18,19)(8,12,22,27)(9,23,25,24)(13,28,29,30)(21,31,32,26)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,51,52,53)(40,55,56,45)(41,57,58,50)(46,61,62,63)(54,64,60,59)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263776);