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 | |
7204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,10)(7,11);(1,2)(3,4)(6,10)(7,11)(13,14,18,19)(15,20,23,27)(16,24,26,17)(21,22,25,28);(5,6)(7,8)(9,10)(11,12)(14,19)(17,24)(20,27)(22,28);(6,10)(7,11)(13,15)(14,20)(16,25)(17,22)(18,23)(19,27)(21,26)(24,28);(5,7,9,11)(6,8,10,12)(13,16)(14,17)(15,21)(18,26)(19,24)(20,28)(22,27)(23,25);(5,8)(6,11)(7,10)(9,12)(13,17)(14,16)(15,22)(18,24)(19,26)(20,25)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72281540 |
| maximal normal subgroup of | #72281540 |
| maximal quotient of | #72282310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7694982);