#99695

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,6,4,7,8,5)(10,14,17,20)(11,18,19,21)(25,26)(27,35,37,39)(28,38,31,32)(29,33)(30,34)(36,40);(2,5)(3,8)(6,7)(9,10,15,14,13,17,12,20)(11,16,18,22,19,24,21,23)(25,27,29,38,30,37,40,32)(26,31,33,39,34,28,36,35);(9,11)(10,16)(12,21)(13,19)(14,22)(15,18)(17,24)(20,23)(25,28,30,31)(26,32,34,38)(27,36,37,33)(29,35,40,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=g,e^2=hi,f^2=gh,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6891);