Finite Group Databasebeta

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Cayley diagram

C2×#11291 #358308

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#11291

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC24C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

1866 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);(4,5)(12,17,20,19)(18,25,26,21);(3,4,6,5)(7,8)(9,10)(11,12)(13,21)(14,19)(15,17)(16,20)(18,23)(22,26)(24,25);(3,5,6,4)(8,10)(11,13,14,22,16,24,15,23)(12,18,17,25,20,26,19,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240061
maximal normal subgroup of#72240061
maximal quotient of#72240039, C2×#121058, #72240041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265504);