Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,28)(17,23)(18,20)(19,32)(21,26)(22,30)(25,40)(27,36)(29,38)(31,37)(33,34)(35,41)(39,42)(43,44);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,20)(21,34)(22,28)(23,32)(24,36)(35,44)(37,38)(39,40);(2,4)(13,15)(14,21)(16,25)(17,26)(18,27)(19,33)(20,34)(22,35)(23,36)(24,32)(28,44)(29,41)(30,42)(31,43)(37,39)(38,40);(6,10)(9,11)(13,16,18,30)(14,22,20,28)(15,25,27,42)(17,31,19,29)(21,35,34,44)(23,38,32,37)(24,39,36,40)(26,43,33,41);(2,4)(7,12)(9,11)(13,17,18,19)(14,23,20,32)(15,26,27,33)(16,29,30,31)(21,36,34,24)(22,37,28,38)(25,41,42,43)(35,39,44,40)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575131);