Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3372 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)(5,6)(7,8)(9,11)(10,12)(14,17)(15,19)(22,26)(24,28);(5,7,10,11)(6,9,12,8)(15,19)(18,20)(21,22,25,26)(23,24,27,28);(5,8,10,9)(6,7,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,27)(22,24,26,28);(1,2,3,4)(5,9,10,8)(6,11,12,7)(13,15)(14,18)(16,19)(17,20)(21,24,25,28)(22,23,26,27);(5,6)(7,9)(8,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72326562, #72326603, #72326616, … |
| maximal normal subgroup of | #72326562, #72326603, #72326616, … |
| maximal quotient of | #72328808, #72332736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375471);