#309006

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 216202);