#6368461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368046
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62590	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1474 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:

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

Polycyclic presentation:

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

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, 6275657);