Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,23)(16,17)(20,28)(21,22)(24,27);(5,6)(7,11)(8,10)(9,12)(13,14,17,19,18,22,16,21)(15,24,25,28,23,20,26,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,26)(17,25)(18,23)(19,27)(21,28)(22,24);(7,12)(9,11)(14,21)(15,23)(16,17)(19,22)(20,27)(24,28);(6,10)(9,11)(13,16,18,17)(14,19,22,21)(15,25,23,26)(20,28,24,27)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72308205, #72308233, #72308503 |
| maximal normal subgroup of | #72308205, #72308233, #72308503 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910404);