#303022

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

717 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)(9,10,14,18)(11,19,22,16)(12,15,17,13)(20,24,23,21)(25,26)(27,34)(28,38)(29,33)(30,32)(31,37)(35,40)(36,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,23)(13,21)(14,22)(15,24)(17,20)(18,19)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(9,12,15,18,14,17,13,10)(11,20,21,19,22,23,24,16)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,17,18,12)(11,21,22,24)(16,20,19,23)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 210218);