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 | |
8672 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)(7,11)(8,9)(10,12)(13,14,18,19)(15,21,22,17)(16,24,25,20)(23,27,28,26);(1,2)(3,4)(6,9)(10,11)(13,15,18,22)(14,17,19,21)(16,23,25,28)(20,27,24,26);(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(16,25)(26,27);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,20)(18,19)(21,22)(23,27)(24,25)(26,28);(5,7)(6,11)(8,12)(9,10)(13,16)(14,20)(15,23)(17,27)(18,25)(19,24)(21,26)(22,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,15)(16,26)(18,21)(19,22)(20,28)(23,24)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72261733, #72262026, #72304527, … |
| maximal normal subgroup of | #72261733, #72262026, #72304527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7873505);