Finite Group Databasebeta

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Cayley diagram

C2×#11524 #480317

order512
nilpotency class4
isoclinismisoclinic class of #11521
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#11524

Subgroups

subgroupG/H
centerC2×C4#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC22#1464
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,7,13,8,12,23,29,9,14,22,30,6,11,20,28)(5,15,18,31,17,10,24,26,16,25,21,33,19,27,32,34);(3,5)(4,10)(6,19)(7,21)(8,17)(9,16)(11,15)(12,25)(13,26)(14,27)(18,22)(20,24)(23,32)(28,31)(29,33)(30,34);(4,11,14,12)(5,16)(7,22)(10,25,27,15)(13,29,30,28)(17,19)(20,23)(26,31,34,33)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal quotient ofC2×#128379, C2×#124849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387513);