#306279

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
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

522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 213475);