Finite Group Databasebeta

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Cayley diagram

C2×#11249 #358101

order512
nilpotency class3
isoclinismisoclinic class of #2018
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11249

Subgroups

subgroupG/H
center, socleC24#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214149, #72216101, C2×#378855
maximal normal subgroup of#72214149, #72216101, C2×#378855
maximal quotient ofC2×#120471

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265297);