Finite Group Databasebeta

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Cayley diagram

C2×#11221 #358095

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11221

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2018 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)(5,6)(8,9)(11,12,14,19,16,20,15,18)(13,17,23,21,24,26,22,25);(7,8,10,9)(11,13)(12,17)(14,23)(15,22)(16,24)(18,25)(19,21)(20,26);(4,6)(7,9,10,8)(11,12)(13,21)(14,19)(15,18)(16,20)(17,22)(23,26)(24,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214159, #72215970, C2×#378848
maximal normal subgroup of#72214159, #72215970, C2×#378848
maximal quotient of#72214111, C2×#120471

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265291);