#6368251

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	#3544	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1443 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^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, 6275447);