Finite Group Databasebeta

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Cayley diagram

C2×#22448 #6379129

order512
nilpotency class3
isoclinismisoclinic class of D4×D8
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#22448

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#135C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265198, #72265213, #72265282, …
maximal normal subgroup of#72265198, #72265213, #72265282, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286325);