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 | |
7463 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)(7,8)(9,10)(11,12)(13,14,19,21)(15,22,16,20)(17,27,28,18)(23,26,25,24);(1,2,3,4)(13,15,19,16)(14,20,21,22)(17,25,28,23)(18,24,27,26);(6,10)(7,11)(14,21)(15,16)(18,27)(23,25);(5,7,9,11)(6,8,10,12)(13,16,19,15)(14,20,21,22)(17,25,28,23)(18,26,27,24);(5,8)(6,11)(7,10)(9,12)(13,17)(14,18)(15,23)(16,25)(19,28)(20,26)(21,27)(22,24);(13,18)(14,17)(15,24)(16,26)(19,27)(20,25)(21,28)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238174, #72238195, #72238220, … |
| maximal normal subgroup of | #72238174, #72238195, #72238220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7652534);