#6356680

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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,16)(13,26)(14,27)(15,21)(17,23)(18,25)(19,24)(22,29)(28,32)(30,31)(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);(1,3,5,15,7,13,6,16)(2,8,20,26,11,21,10,12)(4,14,19,30,18,28,17,29)(9,22,23,32,25,31,24,27)(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)(8,21)(12,26)(14,28)(15,16)(22,31)(27,32)(29,30)(33,36,39,53)(34,41,44,60)(35,46,49,51)(37,52,38,54)(40,55,57,58)(42,59,43,50)(45,61,56,64)(47,62,48,63);(1,4,7,18)(2,9,11,25)(3,14,13,28)(5,19,6,17)(8,22,21,31)(10,24,20,23)(12,27,26,32)(15,30,16,29)(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);(1,5,7,6)(2,10,11,20)(3,15,13,16)(4,17,18,19)(8,12,21,26)(9,23,25,24)(14,29,28,30)(22,32,31,27)(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=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,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, 6263876);