Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,12)(7,16)(9,10)(13,27)(14,26)(15,23)(17,30)(18,19)(20,29)(21,24)(28,36)(32,34);(5,6)(7,13)(8,20)(9,23)(10,12)(11,15)(14,21)(16,25)(17,31)(18,27)(19,33)(22,26)(24,29)(28,30)(32,35)(34,36);(5,7,10,16,11,18,9,19)(6,13,12,25,15,27,23,33)(8,17,21,30,22,32,24,34)(14,28,26,35,29,36,20,31);(5,8)(6,14)(7,17)(9,24)(10,21)(11,22)(12,26)(13,28)(15,29)(16,30)(18,32)(19,34)(20,23)(25,35)(27,36)(31,33);(7,18)(13,27)(16,19)(17,32)(25,33)(28,36)(30,34)(31,35)Polycyclic presentation:
2^9|a^2=h,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72285848, #72285871, … |
| maximal normal subgroup of | #72279230, #72285848, #72285871, … |
| maximal quotient of | #72279630, #72309567, #72322157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287372);