Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,19,23)(15,24,26,28)(16,21,20,27)(17,22,18,25);(5,6)(7,11)(8,10)(9,12)(13,15,18,21,19,26,17,27)(14,20,22,24,23,16,25,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,25)(17,24)(18,28)(19,20)(22,26)(23,27);(7,12)(9,11)(15,26)(16,20)(21,27)(24,28);(6,10)(9,11)(13,17,19,18)(14,22,23,25)(15,27,26,21)(16,28,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281906, #72282514, #72282750 |
| maximal normal subgroup of | #72281906, #72282514, #72282750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413902);