Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#3551 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3188 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)(15,26)(16,31)(17,23)(18,24)(19,25)(20,35)(21,30)(22,32)(27,37)(28,42)(29,39)(33,41)(34,38)(36,43)(40,44);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,28)(17,27)(18,34)(19,30)(20,29)(22,40)(23,42)(24,39)(25,26)(31,37)(32,43)(33,44)(35,38)(36,41);(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,27)(17,19)(18,32)(20,33)(21,36)(23,24)(25,41)(26,40)(28,30)(29,43)(31,35)(34,44)(37,39)(38,42);(7,12)(9,11)(13,17,19,16)(14,23,25,31)(15,28,30,27)(18,32,20,33)(21,37,26,42)(22,35,41,24)(29,43,34,44)(36,38,40,39);(6,10)(9,11)(13,18,19,20)(14,24,25,35)(15,29,30,34)(16,32,17,33)(21,38,26,39)(22,23,41,31)(27,43,28,44)(36,42,40,37)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6407374);