Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628872 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×H(7,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,28)(8,12,19,26)(9,21,23,13)(11,22,25,17)(16,29,30,36)(20,33,32,27)(24,34,35,31);(5,7)(6,11)(8,16)(9,22)(10,18)(12,24)(13,28)(14,25)(15,21)(17,23)(19,30)(20,34)(26,35)(27,36)(29,33)(31,32);(9,23)(13,21)(15,28)(17,22)(20,32)(27,33)(29,36)(31,34);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,f^b=fi,g^b=gi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72227698, #72227731, #72227736, … |
| maximal normal subgroup of | #72227698, #72227731, #72227736, … |
| maximal quotient of | #72262562, #72268006, #72268101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536069);