#304884

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=efh,d^2=eh,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, 212080);