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 | |
8750 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)(7,11)(10,12)(13,14,19,20)(15,17,24,21)(16,22,27,25)(18,26,28,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,20)(15,24)(22,25)(23,26);(6,9)(10,12)(13,15,19,24)(14,21,20,17)(16,26,27,23)(18,25,28,22);(13,16,19,27)(14,22,20,25)(15,23,24,26)(17,18,21,28);(5,6)(7,10)(8,9)(11,12)(13,17,19,21)(14,15,20,24)(16,18,27,28)(22,23,25,26);(5,7)(6,10)(8,11)(9,12)(13,18)(14,23)(15,25)(16,21)(17,27)(19,28)(20,26)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304569, #72304601, #72304639, … |
| maximal normal subgroup of | #72304569, #72304601, #72304639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615960);