Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394020 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,11,18,28)(8,12,19,29)(9,21,24,13)(15,25,33,36)(16,34,35,26)(17,31,22,27)(20,32,23,30);(5,7)(6,11)(8,16)(9,22)(10,18)(12,26)(13,31)(14,28)(15,20)(17,24)(19,35)(21,27)(23,33)(25,30)(29,34)(32,36);(2,4)(5,8)(6,12)(7,15)(9,23)(10,19)(11,25)(13,32)(14,29)(16,22)(17,35)(18,33)(20,24)(21,30)(26,31)(27,34)(28,36);(5,8)(6,12)(7,16)(9,20)(10,19)(11,26)(13,30)(14,29)(15,22)(17,33)(18,35)(21,32)(23,24)(25,31)(27,36)(28,34)Polycyclic presentation:
2^9|a^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251192, #72318110, #72318250, … |
| maximal normal subgroup of | #72251192, #72318110, #72318250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301274);