Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3087 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,22)(19,24)(21,26)(23,27)(25,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,24)(17,23)(19,20)(21,28)(22,27)(25,26);(2,4)(13,15)(14,19)(16,25)(17,23)(18,20)(21,27)(22,28)(24,26);(6,10)(7,12)(14,20)(16,26)(18,19)(21,22)(24,25)(27,28);(15,23)(18,27)(19,28)(24,25)Polycyclic presentation:
2^9|g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72239646, #72249598, #72249607, … |
| maximal normal subgroup of | #72239646, #72249598, #72249607, … |
| maximal quotient of | #72239676, #72264686, #72264692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407905);