Finite Group Databasebeta

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Cayley diagram

C2×#11876 #482045

order512
nilpotency class4
isoclinismisoclinic class of #11757
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#11876

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#2543C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#130280, #72282655, #72282686
maximal normal subgroup ofC2×#130280, #72282655, #72282686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389241);