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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(7,12)(9,11)(13,14,18,21)(15,22,23,19)(16,17,20,27)(24,25,26,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,26)(17,25)(18,23)(20,24)(21,22)(27,28);(5,7,8,12)(6,9,10,11)(15,23)(19,22)(24,26)(25,28);(6,10)(9,11)(13,16)(14,17)(15,24)(18,20)(19,28)(21,27)(22,25)(23,26);(5,7)(6,11)(8,12)(9,10)(13,15)(14,19)(16,26)(17,25)(18,23)(20,24)(21,22)(27,28);(5,6)(7,11)(8,10)(9,12)(13,17)(14,20)(15,25)(16,21)(18,27)(19,24)(22,26)(23,28)Polycyclic presentation:
2^9|a^2=hi,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72300320, #72327640, #72330752, … |
| maximal normal subgroup of | #72300320, #72327640, #72330752, … |
| maximal quotient of | #72300349, #72300423, #72327709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988282);