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 | |
7806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14,18,19)(15,22,20,16)(17,21,26,27)(23,25,28,24);(1,2,3,4)(13,14,18,19)(15,16,20,22)(17,21,26,27)(23,24,28,25);(6,7)(10,11)(14,19)(15,20)(21,27)(23,28);(5,7,9,6)(8,11,12,10)(13,15,18,20)(14,16,19,22)(17,23,26,28)(21,24,27,25);(5,8)(6,10)(7,11)(9,12)(13,16)(14,20)(15,19)(17,25)(18,22)(21,23)(24,26)(27,28);(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,28)(22,25)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72331611, #72336917 |
| maximal normal subgroup of | #72331611, #72336917 |
| maximal quotient of | #72331671, #72332018, #72332089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7850425);