#305218

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=ei,c^2=ef,d^2=eh,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, 212414);