#6355941

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	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1435 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:

(1,2,7,11)(3,12,13,26)(4,9,18,25)(5,10,6,20)(8,15,21,16)(14,27,28,32)(17,23,19,24)(22,30,31,29)(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);(1,3,5,15,7,13,6,16)(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)(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)(8,21)(12,26)(14,28)(15,16)(22,31)(27,32)(29,30)(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)(2,9)(3,14)(5,19)(6,17)(7,18)(8,22)(10,24)(11,25)(12,27)(13,28)(15,30)(16,29)(20,23)(21,31)(26,32)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(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)(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,e^2=h,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, 6263137);