#6368957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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	#3543	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1473 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,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, 6276153);