Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,22)(16,19)(20,27)(21,25)(23,26);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(6,10)(7,11)(13,15,16,22,17,23,19,26)(14,20,18,24,21,27,25,28);(7,11)(9,12);(15,23)(20,27)(22,26)(24,28)Polycyclic presentation:
2^9|b^2=fi,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280255, #72280303, #72283995, … |
| maximal normal subgroup of | #72280255, #72280303, #72283995, … |
| maximal quotient of | #72313063, #72313068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366944);