Finite Group Databasebeta

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Cayley diagram

C2×#11473 #480091

order512
nilpotency class4
isoclinismisoclinic class of #2121
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#11473

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC23C23×C8
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#120C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1612 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);(3,4,6,9,7,10,5,8)(11,12)(13,19)(14,22,23,17)(15,18)(16,21,24,26)(20,25);(11,13)(12,16)(14,21)(15,20)(17,19)(18,24)(22,25)(23,26);(3,5,7,6)(4,8,10,9)(13,20)(16,24)(19,25)(21,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72240312, #72240316, C2×#129599, …
maximal normal subgroup of#72240312, #72240316, C2×#129599, …
maximal quotient of#72261256

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387287);