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 | |
8110 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)(13,14,19,24)(15,20,27,22)(16,25,26,21)(17,18,23,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,28)(17,25)(18,26)(19,27)(21,23)(22,24);(5,7)(6,9)(8,12)(10,11)(13,16,19,26)(14,21,24,25)(15,18,27,28)(17,22,23,20);(13,17,19,23)(14,18,24,28)(15,25,27,21)(16,20,26,22);(5,7)(6,10)(8,12)(9,11)(13,15)(14,22)(16,18)(17,25)(19,27)(20,24)(21,23)(26,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,23)(15,26)(16,27)(17,24)(19,28)(20,21)(22,25)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72279324, C2×#7850169, #72284087, … |
| maximal normal subgroup of | #72279324, C2×#7850169, #72284087, … |
| maximal quotient of | C2×#7850169, #72281005, #72281993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7757365);