#123408

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, prime radical	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=df,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, 30604);