Finite Group Databasebeta

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Cayley diagram

C2×#10457 #240058

order512
nilpotency class3
isoclinismisoclinic class of #10456
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC2×#10457

Subgroups

subgroupG/H
center, socleC23#394
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#3036C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ 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)(5,7)(6,8)(9,10)(11,12)(13,20)(14,22)(15,23)(16,19)(17,21)(18,25)(24,26);(4,7)(8,9)(11,13)(12,17)(14,18)(15,24)(16,22)(19,26)(20,23)(21,25);(4,8)(6,10)(7,9)(11,14,15,17,16,20,25,26)(12,13,18,21,19,22,23,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72213145, #72213372, C2×#240064
maximal normal subgroup of#72213145, #72213372, C2×#240064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147254);