#312072

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

589 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)(9,10,14,15,13,18,12,17)(11,19,22,16,21,24,20,23)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(2,5)(6,7)(10,15)(12,14)(16,19)(17,18)(20,22)(23,24)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40);(9,11,13,21)(10,16,18,23)(12,22,14,20)(15,19,17,24)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(1,3,4,8)(2,6,5,7)(10,15)(12,14)(16,19)(17,18)(20,22)(23,24)

Polycyclic presentation:

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

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, 219268);