#324938

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

525 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)(10,15)(11,18)(12,14)(16,23)(19,20)(22,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(2,5)(6,7)(9,10)(11,16)(12,21)(13,15)(14,17)(18,24)(19,23)(20,22)(25,27,28,34,29,35,36,33)(26,30,32,39,31,38,40,37);(9,11,14,18,13,20,12,19)(10,16,17,24,15,22,21,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,17,15,21)(11,19,20,18)(16,24,22,23)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=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, 232134);