Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622820 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3998 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)(7,12)(8,11)(9,22)(10,13)(14,19)(15,20)(16,26)(17,23)(18,32)(21,24)(25,33)(27,31)(28,30)(29,34)(35,36);(6,11)(8,19)(9,20)(12,23)(13,24)(14,15)(17,31)(18,30)(21,33)(22,25)(26,34)(27,28)(29,36)(32,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,26)(14,27)(15,28)(19,31)(20,30)(21,29)(22,32)(24,34)(25,35)(33,36);(7,16)(12,26)(17,29)(18,30)(23,34)(27,35)(28,32)(31,36);(6,13)(11,24)(12,26)(14,25)(15,22)(23,34)(27,35)(28,32)Polycyclic presentation:
2^9|f^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231499, #72231549, #72231551, … |
| maximal normal subgroup of | #72231499, #72231549, #72231551, … |
| maximal quotient of | #72257068, C2×#7622820, #72309704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530016);