Finite Group Databasebeta

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Cayley diagram

C2×#13093 #494728

order512
nilpotency class4
isoclinismisoclinic class of #11549
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#13093

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#2431C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

2326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#496222, #72340704, #72340705
maximal normal subgroup ofC2×#496222, #72340704, #72340705

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401924);