#383234

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1887 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=dei,f^c=fi,e^d=ei,f^d=fi

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, 290430);