#389115

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1245 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)(5,6,8,7)(9,10,15,21)(11,22,25,36)(12,26,13,18)(14,31,16,20)(17,32,35,24)(19,30,37,28)(23,39,40,34)(27,33,29,38);(9,11,14,24,15,25,16,32)(10,17,20,22,21,35,31,36)(12,27,28,23,13,29,30,40)(18,34,19,38,26,39,37,33);(2,4)(5,7,8,6)(9,12,15,13)(10,18,21,26)(11,23,25,40)(14,28,16,30)(17,33,35,38)(19,31,37,20)(22,34,36,39)(24,29,32,27);(2,4)(9,13,15,12)(10,19,21,37)(11,23,25,40)(14,28,16,30)(17,34,35,39)(18,31,26,20)(22,33,36,38)(24,27,32,29)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei

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, 296311);