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 | |
7836 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)(15,17)(16,19)(18,21)(20,22)(23,25)(24,27)(26,28);(1,2)(3,4)(13,15)(14,17)(16,22)(18,23)(19,20)(21,25)(24,26)(27,28);(6,7)(10,11)(14,16)(15,20)(21,24)(23,28);(5,7,9,6)(8,11,12,10)(13,16,19,14)(15,22,20,17)(18,24,27,21)(23,26,28,25);(5,8,9,12)(6,10,7,11)(13,17,19,22)(14,20,16,15)(18,26,27,25)(21,23,24,28);(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,28)(22,26)Polycyclic presentation:
2^9|a^2=hi,d^2=h,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 | #72236855, #72238143, #72238156, … |
| maximal normal subgroup of | #72236855, #72238143, #72238156, … |
| maximal quotient of | #72330890, #72331698, #72331872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825085);