#6357956

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265152);