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 |
| prime radical | #24428 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1897 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,25)(15,26,16,22)(17,23,33,42)(18,24,19,34)(21,30,37,28)(27,39,29,38)(31,36,43,44)(32,40,35,41);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,25,37)(17,31,33,43)(18,28,19,30)(22,34,26,24)(23,39,42,38)(27,32,29,35)(36,41,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,15)(14,22,25,26)(17,27,33,29)(18,28,19,30)(21,34,37,24)(23,39,42,38)(31,32,43,35)(36,40,44,41);(7,12)(9,11)(13,17,20,33)(14,23,25,42)(15,27,16,29)(18,32,19,35)(21,36,37,44)(22,38,26,39)(24,40,34,41)(28,31,30,43);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,28,16,30)(17,32,33,35)(21,22,37,26)(23,40,42,41)(27,31,29,43)(36,38,44,39)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=hi,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=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970218);