#383224

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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

1878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290420);