#403672

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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)(11,20)(12,25,27,31)(13,18)(14,19)(15,23)(16,26,32,35)(17,21)(22,28,37,39)(24,29)(30,34)(33,36,38,40);(1,2,4,6)(3,7,8,5)(9,11,21,34,14,24,13,23)(10,15,18,29,19,30,17,20)(12,22,35,40,27,37,26,36)(16,28,25,38,32,39,31,33);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,17)(14,19)(18,21)(20,23)(22,28)(24,30)(25,35)(26,31)(27,32)(29,34)(33,36)(37,39)(38,40);(9,12,13,26,14,27,21,35)(10,16,17,31,19,32,18,25)(11,22,23,36,24,37,34,40)(15,28,20,33,30,39,29,38)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,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, 310868);