#304965

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

667 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)(9,10,13,16)(11,17,19,15)(12,14,21,22)(18,20,24,23)(25,26)(27,30)(28,31)(29,37)(32,36)(33,38)(34,40)(35,39);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,16,17)(12,18,21,24)(14,23,22,20)(25,27)(26,30)(28,35)(29,33)(31,39)(32,40)(34,36)(37,38);(1,3,5,4)(2,7,8,6)(9,12,14,16,13,21,22,10)(11,18,20,15,19,24,23,17)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(1,4,5,3)(2,7,8,6)(10,16)(12,21)(15,17)(18,24)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,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=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, 212161);