#416540

order	512
nilpotency class	3
isoclinism	isoclinic class of #397240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fh,c^2=e,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323736);