#324970

order	512
nilpotency class	3
isoclinism	isoclinic class of #278987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=f,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dh,d^c=dh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232166);