#304584

order	512
nilpotency class	3
isoclinism	isoclinic class of #271515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

745 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:

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

Polycyclic presentation:

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

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, 211780);