#290365

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

549 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,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, 197561);