Finite Group Databasebeta

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Cayley diagram

C2×#11891 #482086

order512
nilpotency class4
isoclinismisoclinic class of #11855
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11891

Subgroups

subgroupG/H
center, socleC23C2×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#466C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482090
maximal normal subgroup ofC2×#482090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389282);