Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2834 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)(13,14,19,16)(15,22,26,28)(17,21,18,27)(20,25,23,24);(5,6)(7,10)(8,9)(11,12)(13,15,18,24,19,26,17,25)(14,20,21,22,16,23,27,28);(1,2,3,4)(6,9)(10,12)(13,16,19,14)(15,23,26,20)(17,21,18,27)(22,24,28,25);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,21,16,27)(15,24,26,25)(20,28,23,22);(7,11)(10,12)(13,17,19,18)(14,21,16,27)(15,25,26,24)(20,22,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307388, #72307449, #72307590, … |
| maximal normal subgroup of | #72307388, #72307449, #72307590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465316);