Finite Group Databasebeta

← Prev (#181191) Next (#181193) →

Cayley diagram

C2×#7967 #181192

order512
nilpotency class3
isoclinismisoclinic class of #7943
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#7967

Subgroups

subgroupG/H
center, socleC23C2×#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4,6,11,8,12,7,10)(5,13,15,9,14,18,16,17)(19,20)(21,27,29,33)(22,25)(23,26)(24,28,31,34)(30,32);(3,5,7,16,8,14,6,15)(4,9,11,18,12,17,10,13)(19,21)(20,24,26,31)(22,28)(23,29)(25,27,32,33)(30,34);(5,14)(9,17)(13,18)(15,16)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#202075, C2×#181200
maximal normal subgroup ofC2×#202075, C2×#181200
maximal quotient of#72309156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88388);