Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,23)(15,24,21,17)(16,20,18,22)(25,28,26,27);(1,2,3,4)(5,7,9,8)(6,10,11,12)(15,21)(17,24)(25,26)(27,28);(7,8)(10,12)(13,15,19,21)(14,17,23,24)(16,26,18,25)(20,28,22,27);(13,16,19,18)(14,20,23,22)(15,25,21,26)(17,27,24,28);(5,6)(7,12)(8,10)(9,11)(13,17)(14,21)(15,23)(16,27)(18,28)(19,24)(20,26)(22,25);(5,8,9,7)(6,10,11,12)(13,18,19,16)(14,22,23,20)(15,26,21,25)(17,28,24,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280361, #72280484, #72282475, … |
| maximal normal subgroup of | #72280361, #72280484, #72282475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7653946);