#304902

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

726 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)(11,17)(12,21)(13,16)(14,22)(15,19)(18,20)(23,24)(25,26,30,34)(27,36,37,32)(28,39,38,40)(29,35,31,33);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,16,17)(12,18,22,23)(14,20,21,24)(25,27,30,37)(26,32,34,36)(28,29,38,31)(33,40,35,39);(1,3,5,4)(2,7,8,6)(9,12,21,10,13,22,14,16)(11,18,24,15,19,23,20,17)(25,28,31,27,30,38,29,37)(26,32,35,40,34,36,33,39);(1,4,5,3)(2,7,8,6)(10,16)(11,19)(12,22)(20,24)(25,29,30,31)(26,33,34,35)(27,28,37,38)(32,39,36,40)

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=ef,d^2=ehi,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, 212098);