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 | |
9392 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)(6,9)(7,11)(14,19)(15,20)(17,25)(23,28);(1,2)(3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,16,20,24)(17,21,25,26)(22,23,27,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,27)(26,28);(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,26)(25,28);(13,17)(14,21)(15,22)(16,23)(18,25)(19,26)(20,27)(24,28)Polycyclic presentation:
2^9|c^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72261660, #72262082, #72312926, … |
| maximal normal subgroup of | #72261660, #72262082, #72312926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552500);