Finite Group Databasebeta

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Cayley diagram

C2×#18176 #6346432

order512
nilpotency class3
isoclinismisoclinic class of #18162
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#18176

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#1595C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

3352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346643, C2×#6346750, C2×#6346716
maximal normal subgroup ofC2×#6346643, C2×#6346750, C2×#6346716
maximal quotient ofC2×#208860, C2×#241504, #72309331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253628);