Finite Group Databasebeta

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Cayley diagram

C2×#33436 #7622573

order512
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C24×C8 · Schur multiplier (H2) = C29

Aliases

direct productC2×#33436

Subgroups

subgroupG/H
centerC22×C8C24
commutatorC22C24×C8
Frattini, cosocleC2×C8C25
socleC23C23×C8
prime radicalC2×#525C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(3,4)(5,6)(7,8,10,14,11,15,20,26,12,16,21,27,22,28,32,36)(9,13,18,24,19,25,31,35,17,23,29,33,30,34,37,38);(4,6);(7,9,10,18,11,19,20,31,12,17,21,29,22,30,32,37)(8,13,14,24,15,25,26,35,16,23,27,33,28,34,36,38);(9,17)(13,23)(18,29)(19,30)(24,33)(25,34)(31,37)(35,38)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D4×#504
maximal normal subgroup ofC2×D4×#504
maximal quotient ofC2×#483602, #72322022, #72322081

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529769);