#403403

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:

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

Polycyclic presentation:

2^9|a^2=e,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, 310599);