Finite Group Databasebeta

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Cayley diagram

C2×#11797 #481812

order512
nilpotency class4
isoclinismisoclinic class of #11796
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#11797

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,19)(14,22,23,17)(15,18)(16,20,24,26)(21,25);(4,7)(5,10)(8,9)(11,13,15,21)(12,16,18,24)(14,20,23,26)(17,19,22,25);(4,8)(7,9)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#482360, #72240374, #72240381, …
maximal normal subgroup ofC2×#482360, #72240374, #72240381, …
maximal quotient of#72261849, #72293203, #72293207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389008);