#6355938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
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	#3511	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1449 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)(35,39)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57)(44,53)(45,61)(48,56)(50,55)(52,59)(54,60)(58,64)(62,63);(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)(34,39)(36,43)(37,45)(38,44)(40,52)(41,53)(42,54)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(3,13)(7,20)(11,24)(14,27)(15,19)(21,31)(25,32)(28,30)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(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)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(4,16)(8,22)(14,27)(17,26)(21,31)(23,29)(25,32)(28,30)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=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, 6263134);