#389121

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1271 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,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, 296317);