Finite Group Databasebeta

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Cayley diagram

C2×#10998 #357550

order512
nilpotency class3
isoclinismisoclinic class of #2002
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#10998

Subgroups

subgroupG/H
center, socleC24#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

4630 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);(7,8,9,10)(12,16)(14,21)(17,25)(18,22)(19,23)(24,26);(3,4)(5,6)(11,12)(13,17)(14,22)(15,18)(16,21)(19,24)(20,26)(23,25);(4,6)(11,13,15,20)(12,17,18,26)(14,23,21,19)(16,24,22,25)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#357552, #72317090, C2×#394095
maximal normal subgroup ofC2×#357552, #72317090, C2×#394095

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264746);