Finite Group Databasebeta

← Prev (#6446707) Next (#6446709) →

Cayley diagram

C2×#25875 #6446708

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25875

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3498C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312145, C2×#6487513
maximal normal subgroup of#72312145, C2×#6487513

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353904);