#258994

order	512
nilpotency class	3
isoclinism	isoclinic class of #246415
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

747 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)(3,7)(4,5)(6,8)(9,10,13,17)(11,19,12,15)(14,23,22,18)(16,20,24,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(2,5)(6,7)(9,11,13,12)(10,15,17,19)(14,21,22,20)(16,23,24,18)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(9,12,13,11)(10,16,17,24)(14,20,22,21)(15,18,19,23)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,24)(18,23)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=ehi,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,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166190);