Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1265 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11)(8,18)(9,21)(10,13)(12,19)(14,24)(15,31)(16,29)(17,26)(20,30)(22,27)(23,28)(25,32)(33,36)(34,35);(1,2)(3,4)(5,7,10,17)(6,11,13,26)(8,15,20,33)(9,22,24,16)(12,25,28,35)(14,29,21,27)(18,31,30,36)(19,32,23,34);(1,3,4,2)(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,28)(14,30)(16,34)(17,33)(18,21)(19,24)(22,32)(26,35)(27,36)(29,31);(6,13)(11,26)(12,28)(14,21)(18,30)(25,35)(27,29)(31,36)Polycyclic presentation:
2^9|b^2=i,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,f^c=fi,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286056, #72286158, #72306898, … |
| maximal normal subgroup of | #72286056, #72286158, #72306898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406334);