#206036

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi

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, 113232);