#383228

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
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

1877 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)(11,19)(12,24)(13,27)(17,25)(18,31)(21,35)(22,23)(29,33)(30,34)(37,40)(38,39);(1,2)(3,4)(9,10,13,15,14,18,27,31)(11,20,22,33,23,16,19,29)(12,25,26,34,24,30,28,17)(21,37,35,32,39,40,38,36);(2,4)(9,11)(10,16)(12,21)(13,22)(14,23)(15,29)(17,32)(18,20)(19,27)(24,39)(25,40)(26,35)(28,38)(30,37)(31,33)(34,36);(9,12,14,24)(10,17,18,34)(11,21,23,39)(13,28,27,26)(15,30,31,25)(16,32,20,36)(19,35,22,38)(29,37,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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=de,f^c=fi,e^d=ei,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, 290424);