#287493

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 194689);