Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3221 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)(13,14,17,23,19,25,18,24)(15,21,27,20,26,28,16,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,25,22)(16,18,27,17)(21,24,28,23);(6,9)(10,12)(13,16)(14,21)(15,18)(17,26)(19,27)(20,24)(22,23)(25,28);(6,9)(10,12)(13,16)(14,22)(15,17)(18,26)(19,27)(20,25)(21,23)(24,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,16,26,27)(20,21,22,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256854, #72256959, #72257082, … |
| maximal normal subgroup of | #72256854, #72256959, #72257082, … |
| maximal quotient of | #72264233, #72264309, #72308927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398751);