#123410

order	512
nilpotency class	4
isoclinism	isoclinic class of #123403
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=dfhi,c^2=egh,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30606);