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 | |
7507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,19,16)(15,17,22,20)(18,21,27,24)(23,25,28,26);(1,2,3,4)(13,15,19,22)(14,20,16,17)(18,23,27,28)(21,26,24,25);(6,10)(7,11)(14,16)(15,22)(21,24)(23,28);(5,7,9,11)(6,8,10,12)(13,16,19,14)(15,17,22,20)(18,24,27,21)(23,25,28,26);(5,8)(6,11)(7,10)(9,12)(13,17)(14,15)(16,22)(18,26)(19,20)(21,28)(23,24)(25,27);(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238194, #72238195, #72238210, … |
| maximal normal subgroup of | #72238194, #72238195, #72238210, … |
| maximal quotient of | #72263363, #72263832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759595);