Finite Group Databasebeta

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Cayley diagram

C2×#17649 #6343754

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#17649

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC23C22×C22⋊C4
prime radicalC2×#3444C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

3478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(4,8,9,10)(11,12,15,18,16,19,14,23)(13,20,21,25,22,17,24,26);(3,4)(5,8)(6,10)(7,9)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,20)(23,26);(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,20,26);(3,5,7,6)(4,8,9,10)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#186568, #72239500, #72239501
maximal normal subgroup ofC2×#186568, #72239500, #72239501
maximal quotient of#72239496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250950);