#393905

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
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
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3941 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)(11,20)(12,25)(13,30)(14,18)(15,19)(16,23)(17,27)(21,31)(22,35)(24,32)(26,33)(28,34)(29,36)(37,39)(38,40);(2,4)(9,11)(10,16)(12,26)(13,24)(14,23)(15,22)(17,33)(18,35)(19,32)(20,30)(21,28)(25,31)(27,38)(29,39)(34,37)(36,40);(9,12,15,29)(10,17,19,34)(11,21,22,38)(13,28,14,27)(16,31,32,40)(18,36,30,25)(20,37,35,33)(23,26,24,39);(6,8)(11,22)(16,32)(20,35)(21,38)(23,24)(26,39)(31,40)(33,37)

Polycyclic presentation:

2^9|c^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,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301101);