Finite Group Databasebeta

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Cayley diagram

C2×#17578 #6343350

order512
nilpotency class3
isoclinismisoclinic class of #17577
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#17578

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC26D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

5002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72238875, #72238876, #72238877, …
maximal normal subgroup of#72238875, #72238876, #72238877, …
maximal quotient of#72238892, C2×#206983, C2×#189840, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250546);