#312536

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
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

560 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)(9,10,14,15,13,18,12,17)(11,19,22,16,21,23,20,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,36,40,39);(2,5)(6,7)(10,15)(12,14)(16,19)(17,18)(20,22)(23,24)(25,27,29,34)(26,31,30,38)(28,36,37,39)(32,33,35,40);(9,11,13,21)(10,16,18,24)(12,22,14,20)(15,19,17,23)(25,28,29,37)(26,32,30,35)(27,33,34,40)(31,39,38,36);(1,3,4,8)(2,6,5,7)(10,17)(12,14)(15,18)(16,23)(19,24)(20,22)

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=hi,d^2=efhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219732);