#304954

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212150);