Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(2,4)(6,10)(7,12)(13,14,19,20)(15,21,24,16)(17,22,31,34)(18,32,33,23)(25,29,36,27)(26,28,38,35)(30,42,43,37)(39,40,44,41);(5,6)(7,11)(8,12)(9,10)(14,20)(15,24)(22,34)(23,32)(25,36)(26,38)(37,42)(39,44);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,21)(16,20)(17,25)(18,26)(19,24)(22,29)(23,35)(27,34)(28,32)(30,39)(31,36)(33,38)(37,41)(40,42)(43,44);(5,8,9,11)(6,12,10,7)(13,16,19,21)(14,15,20,24)(17,29,31,27)(18,28,33,35)(22,36,34,25)(23,26,32,38)(30,41,43,40)(37,44,42,39);(13,17)(14,22)(15,25)(16,27)(18,30)(19,31)(20,34)(21,29)(23,37)(24,36)(26,39)(28,40)(32,42)(33,43)(35,41)(38,44);(13,18,19,33)(14,23,20,32)(15,26,24,38)(16,28,21,35)(17,30,31,43)(22,37,34,42)(25,39,36,44)(27,40,29,41)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538129);