#309208

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
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

521 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)(11,17)(12,22)(13,16)(14,24)(15,20)(18,21)(19,23)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(2,5)(6,7)(9,11,14,13,15,21,23,12)(10,16,19,17,20,22,24,18)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(9,12,14,11,15,13,23,21)(10,17,19,22,20,18,24,16)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,8)(2,6,5,7)(9,13,14,21,15,12,23,11)(10,18,19,16,20,17,24,22)

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,d^b=dh,g^b=gi,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, 216404);