Finite Group Databasebeta

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Cayley diagram

C2×#12883 #494070

order512
nilpotency class3
isoclinismisoclinic class of #465
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#12883

Subgroups

subgroupG/H
centerC22×C8C2×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC23C2×C4⋊C8
prime radicalC24C2×M16
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

830 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)(5,8)(6,7)(9,10)(11,12,15,20,16,19,24,26,17,21,23,18,14,25,13,22);(4,7)(8,10)(11,13,14,23,17,24,16,15)(12,18,19,22,21,20,25,26);(3,5,6,9)(4,8,7,10)(11,13,14,23,17,24,16,15)(12,18,19,22,21,20,25,26)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#135732, #72210317
maximal normal subgroup ofC2×#135732, #72210317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401266);