#393101

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

1788 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)(9,10,16,20,14,22,15,21)(11,23,25,36,27,17,26,37)(12,28,24,39,29,38,30,18)(13,31,32,19,34,40,33,35);(1,2,3,4)(9,11,16,26,14,27,15,25)(10,17,21,37,22,23,20,36)(12,13,24,33,29,34,30,32)(18,35,38,31,39,19,28,40);(9,12,16,24,14,29,15,30)(10,18,21,38,22,39,20,28)(11,13,26,33,27,34,25,32)(17,35,37,31,23,19,36,40);(6,8)(9,13)(10,19)(11,24)(12,25)(14,34)(15,32)(16,33)(17,28)(18,37)(20,31)(21,40)(22,35)(23,38)(26,29)(27,30)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,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, 300297);