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 | |
8491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,20,21,16)(17,22,26,27)(23,28,25,24);(1,2)(3,4)(13,15)(14,16)(17,25)(18,21)(19,20)(22,28)(23,26)(24,27);(6,9)(10,11)(14,19)(15,21)(17,26)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,28)(24,27)(25,26);(5,7)(6,10)(8,12)(9,11)(13,16,18,20)(14,21,19,15)(17,24,26,28)(22,25,27,23);(13,17,18,26)(14,22,19,27)(15,23,21,25)(16,24,20,28)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72236869, #72238160, #72238199, … |
| maximal normal subgroup of | #72236869, #72238160, #72238199, … |
| maximal quotient of | #72280697, #72280893, #72281037 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7753960);