Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14,19,20,18,24,17,23)(15,25,31,21,28,38,27,35)(16,22,32,33,29,36,30,37)(26,39,42,34,40,44,41,43);(2,4)(5,6)(7,10)(8,9)(11,12)(14,20)(17,19)(21,25)(22,33)(23,24)(27,31)(30,32)(34,39)(35,38)(36,37)(41,42)(43,44);(6,9)(7,11)(13,15,18,28)(14,21,24,35)(16,26,29,40)(17,31,19,27)(20,25,23,38)(22,34,36,43)(30,42,32,41)(33,39,37,44);(5,7,8,11)(6,10,9,12)(13,16,18,29)(14,22,24,36)(15,26,28,40)(17,30,19,32)(20,33,23,37)(21,34,35,43)(25,39,38,44)(27,41,31,42);(6,9)(10,12)(16,29)(22,36)(26,40)(30,32)(33,37)(34,43)(39,44)(41,42)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420514);