#297001

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

705 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)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,31,34)(27,32,36,39)(28,37,35,40)(29,33,30,38);(2,5)(6,7)(9,10,13,15)(11,14,18,21)(12,16,19,22)(17,20,23,24)(25,27,31,36)(26,32,34,39)(28,29,35,30)(33,40,38,37);(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(25,28,30,27,31,35,29,36)(26,32,33,37,34,39,38,40);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,22)(17,23)(25,29,31,30)(26,33,34,38)(27,28,36,35)(32,37,39,40)

Polycyclic presentation:

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

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, 204197);