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 | |
8918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14,19,24)(15,25,26,20)(16,17,22,21)(18,23,28,27);(1,2)(3,4)(13,15)(14,20)(16,18)(17,27)(19,26)(21,23)(22,28)(24,25);(6,10)(7,11)(15,26)(16,22)(17,21)(20,25);(5,7,9,11)(6,8,10,12)(13,16,19,22)(14,21,24,17)(15,18,26,28)(20,23,25,27);(5,8)(6,11)(7,10)(9,12)(13,17,19,21)(14,22,24,16)(15,27,26,23)(18,20,28,25);(13,18)(14,23)(15,22)(16,26)(17,20)(19,28)(21,25)(24,27)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256602, #72256673, #72257999, … |
| maximal normal subgroup of | #72256602, #72256673, #72257999, … |
| maximal quotient of | #72256744, #72258899, #72263442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650392);