#393903

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301099);