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 | |
9176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(6,9)(10,11)(13,14)(15,22)(16,18)(17,21)(19,20)(23,28)(24,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,23)(20,22)(21,26)(24,25)(27,28);(2,4)(5,6)(7,11)(8,9)(10,12)(14,20)(15,16)(17,25)(26,28);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16,19,15)(14,18,20,22)(17,24,25,23)(21,26,27,28);(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28);(13,18)(14,15)(16,20)(17,26)(19,22)(21,23)(24,27)(25,28)Polycyclic presentation:
2^9|d^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72256612, #72256695, #72256696, … |
| maximal normal subgroup of | #72256612, #72256695, #72256696, … |
| maximal quotient of | #72263410, #72263412, #72263544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7579976);