#243557

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
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

615 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)(3,6)(4,5)(7,8)(9,10,13,17,12,16,21,15)(11,18,20,23,19,14,22,24)(25,26)(27,35)(28,33)(29,32)(30,34)(31,38)(36,39)(37,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,20)(15,23)(16,18)(17,24)(21,22)(25,27,29,38)(26,31,32,35)(28,37,30,36)(33,39,34,40);(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)(26,32)(27,36)(28,30)(31,39)(35,40)(37,38);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,17)(20,22)(25,28,29,30)(26,33,32,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150753);