Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,17,19,22,27,28)(15,21,24,23,16,25,26,20);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,19)(17,25)(18,26)(21,28)(22,23)(24,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,15)(14,21,22,25)(17,23,28,20)(18,26,27,24);(7,12)(9,11)(13,17,27,14,19,28,18,22)(15,23,26,21,16,20,24,25);(6,10)(9,11)(15,16)(20,23)(21,25)(24,26)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251266, #72251272, #72256960, … |
| maximal normal subgroup of | #72251266, #72251272, #72256960, … |
| maximal quotient of | #72265071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928781);