Finite Group Databasebeta

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Cayley diagram

C2×#11884 #482058

order512
nilpotency class4
isoclinismisoclinic class of #11757
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#11884

Subgroups

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

1432 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)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(4,8)(5,6)(9,10)(11,13)(12,17)(14,24)(15,22)(16,23)(18,25)(19,21)(20,26);(3,5,7,6)(4,8,10,9)(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282573, #72282655, #72282675
maximal normal subgroup of#72282573, #72282655, #72282675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389254);