Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,16)(17,21)(18,20)(19,22)(23,24)(25,28)(26,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(2,4)(14,20)(15,22)(21,27)(23,28);(1,2)(3,4)(13,15)(14,19)(16,20)(17,23)(18,22)(21,25)(24,27)(26,28);(5,7,8,11)(6,9,10,12)(13,16,18,19)(14,15,20,22)(17,25,26,24)(21,28,27,23);(13,17)(14,21)(15,23)(16,24)(18,26)(19,25)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72238183, #72238186, #72238207, … |
| maximal normal subgroup of | #72238183, #72238186, #72238207, … |
| maximal quotient of | #72313384, #72313640, #72313864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603200);