#403404

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2238 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:

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

Polycyclic presentation:

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

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, 310600);