#304991

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

681 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)(9,10,13,17)(11,19,20,16)(12,15,18,14)(21,24,23,22)(25,26,30,34)(27,35,32,31)(28,37,39,38)(29,36,40,33);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,22)(13,20)(14,21)(15,23)(17,19)(18,24)(25,27)(26,31)(28,36)(29,37)(30,32)(33,39)(34,35)(38,40);(1,3,5,4)(2,7,8,6)(9,12,14,10,13,18,15,17)(11,16,21,24,20,19,23,22)(25,28,29,35,30,39,40,31)(26,32,33,37,34,27,36,38);(1,4,5,3)(2,7,8,6)(10,17)(12,18)(16,19)(22,24)(25,29,30,40)(26,33,34,36)(27,37,32,38)(28,31,39,35)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=efh,d^2=ehi,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, 212187);