Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622820 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,22)(10,13)(14,19)(15,20)(16,24)(17,23)(18,32)(21,27)(25,31)(26,30)(28,33)(29,34)(35,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,23)(13,24)(14,25)(15,26)(19,31)(20,30)(21,29)(22,32)(27,34)(28,35)(33,36);(6,12)(8,19)(9,20)(11,23)(13,27)(14,15)(17,31)(18,30)(21,33)(22,28)(24,34)(25,26)(29,36)(32,35);(7,16)(11,24)(17,29)(18,30)(23,34)(25,35)(26,32)(31,36);(6,13)(11,24)(12,27)(14,28)(15,22)(23,34)(25,35)(26,32)Polycyclic presentation:
2^9|b^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=di,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237272, #72239538, #72239554, … |
| maximal normal subgroup of | #72237272, #72239538, #72239554, … |
| maximal quotient of | #72319072, #72319972, C2×#7622935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530131);