#303937

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

743 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,14)(13,17)(15,18)(16,20)(21,24)(22,23)(25,26,31,34)(27,32,36,38)(28,35,39,40)(29,33,30,37);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,16,17,19)(12,22,18,24)(14,21,15,23)(25,27)(26,32)(28,37)(29,35)(30,40)(31,36)(33,39)(34,38);(1,3,5,4)(2,7,8,6)(9,12,15,17,13,18,14,10)(11,16,23,22,20,19,21,24)(25,28,30,38,31,39,29,32)(26,27,33,35,34,36,37,40);(1,4,5,3)(2,7,8,6)(10,17)(12,18)(16,19)(22,24)(25,29,31,30)(26,33,34,37)(27,35,36,40)(28,32,39,38)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ef,d^2=eh,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211133);