#304977

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, 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

657 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)(11,18)(12,22)(13,16)(14,17)(15,21)(19,20)(23,24)(25,26,29,32)(27,34,36,31)(28,37,39,40)(30,35,38,33);(1,3,5,4)(2,6,8,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,20)(23,24)(25,27,29,36)(26,31,32,34)(28,35,39,33)(30,40,38,37);(1,3,5,4)(2,7,8,6)(9,11,12,21,14,20,13,23)(10,15,16,18,17,24,22,19)(25,28,38,34,29,39,30,31)(26,27,35,40,32,36,33,37);(1,4,5,3)(2,7,8,6)(11,20)(15,24)(18,19)(21,23)(26,32)(28,39)(31,34)(33,35)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=ei,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, 212173);