Finite Group Databasebeta

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Cayley diagram

C2×#8388 #185706

order512
nilpotency class3
isoclinismisoclinic class of #8359
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8388

Subgroups

subgroupG/H
center, socleC23#379
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

3384 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);(3,4,6,8,7,10,5,9)(11,12,15,20,16,19,23,26)(13,18,21,25,14,17,22,24);(4,8)(5,6)(9,10)(11,13,16,14)(12,17,19,18)(15,22,23,21)(20,25,26,24);(11,14,16,13)(12,18,19,17)(15,21,23,22)(20,24,26,25)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72257396, C2×#185759
maximal normal subgroup of#72257396, C2×#185759
maximal quotient of#72264381

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92902);