#393070

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300266);