#305099

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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

731 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)(11,21,17,16)(12,23,24,22)(13,20,15,18)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,18)(13,22)(14,17)(15,23)(19,21)(20,24)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40);(1,3,5,4)(2,7,8,6)(9,12,13,16,14,24,15,21)(10,17,18,22,19,11,20,23)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,18,19,20)(11,22,17,23)(12,21,24,16)

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,c^2=ef,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=dh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212295);