Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3113 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)(5,6,10,17,11,15,27,16)(7,18,21,30,19,35,20,12)(8,13,23,31,22,32,36,29)(9,14,25,34,26,33,24,28);(5,7,10,21,11,19,27,20)(6,12,17,18,15,30,16,35)(8,9,23,25,22,26,36,24)(13,28,31,14,32,34,29,33);(2,4)(5,8)(6,13)(7,9)(10,23)(11,22)(12,28)(14,18)(15,32)(16,29)(17,31)(19,26)(20,24)(21,25)(27,36)(30,34)(33,35);(5,9)(6,14)(7,8)(10,25)(11,26)(12,29)(13,18)(15,33)(16,28)(17,34)(19,22)(20,36)(21,23)(24,27)(30,31)(32,35);(6,15)(7,19)(8,22)(14,33)(16,17)(20,21)(23,36)(28,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72320787 |
| maximal normal subgroup of | #72320787 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288696);