#305247

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212443);