Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4434 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)(6,7)(8,12)(14,17)(15,19)(21,22)(23,27)(24,28)(25,26);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,27,28,24);(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,28)(26,27);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,26)(27,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|b^2=f,d^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72328559, #72328577, #72328702, … |
| maximal normal subgroup of | #72328559, #72328577, #72328702, … |
| maximal quotient of | #72330682, #72332538 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 782498);