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 | |
8100 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)(13,14,19,21)(15,23,24,20)(16,26,27,18)(17,28,25,22);(1,2,3,4)(13,15,19,24)(14,20,21,23)(16,25,27,17)(18,22,26,28);(6,7)(10,11)(14,21)(15,24)(16,27)(22,28);(5,7,9,6)(8,11,12,10)(13,16)(14,18)(15,25)(17,24)(19,27)(20,22)(21,26)(23,28);(5,8,9,12)(6,10,7,11)(13,17,19,25)(14,22,21,28)(15,16,24,27)(18,20,26,23);(13,18)(14,16)(15,22)(17,23)(19,26)(20,25)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=hi,e^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 | #72326661 |
| maximal normal subgroup of | #72326661 |
| maximal quotient of | #72326670, #72326737, #72326966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7802117);