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 | |
3264 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)(13,14,16,25,18,22,17,21)(15,20,26,19,24,27,23,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,22,28)(16,23,17,26)(20,21,27,25);(6,9)(10,12)(13,15)(14,19)(16,23)(17,26)(18,24)(20,25)(21,27)(22,28);(6,9)(10,12)(13,15)(14,20)(16,26)(17,23)(18,24)(19,25)(21,28)(22,27);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,22,25)(15,23,24,26)(19,27,28,20)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,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 | #72256890, #72257027, #72257048, … |
| maximal normal subgroup of | #72256890, #72257027, #72257048, … |
| maximal quotient of | #72264309, #72264576, #72265061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398718);