Finite Group Databasebeta

← Prev (#126604) Next (#126606) →

Cayley diagram

C2×#7146 #126605

order512
nilpotency class3
isoclinismisoclinic class of #1284
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#7146

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C2×C4×C8
Frattini, cosocleC2×C4×C8C23
socleC23#336
prime radicalC24C4⋊C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72210666, #72210669
maximal normal subgroup of#72210666, #72210669
maximal quotient ofC2×#93250, C2×#94008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33801);