Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,26,21)(16,22,27,28)(17,23,18,20);(5,6)(7,11)(8,10)(9,12)(14,20)(15,26)(16,25)(17,18)(21,27)(23,24);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,26)(14,21,24,25)(16,23,27,20)(17,28,18,22);(7,12)(9,11)(13,16,19,27)(14,22,24,28)(15,20,26,23)(17,21,18,25);(6,10)(9,11)(13,17,19,18)(14,20,24,23)(15,22,26,28)(16,21,27,25)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283384, #72283676 |
| maximal normal subgroup of | #72283384, #72283676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970062);