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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 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)(7,11)(10,12)(14,18)(15,22)(16,24)(19,26)(20,21)(25,27);(6,9)(7,11)(13,14,16,18,17,21,24,20)(15,19,23,25,22,27,28,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,22)(14,19,21,27)(16,23,24,28)(18,25,20,26);(6,9)(10,12)(14,20)(16,24)(18,21)(19,26)(23,28)(25,27);(5,7,8,11)(6,10,9,12)(15,22)(19,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=i,e^2=h,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,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72308151, #72308233, #72308473 |
| maximal normal subgroup of | #72308151, #72308233, #72308473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056495);