Finite Group Databasebeta

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Cayley diagram

C2×#32026 #7507478

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#32026

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2838 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)(5,7)(6,8)(9,10)(11,12)(13,17)(14,16)(15,18);(3,5,7,4)(6,9,10,8)(11,13,14,18)(12,15,16,17)(19,20)(21,23)(22,25)(24,26);(3,6,7,10)(4,8,5,9)(12,16)(15,17)(20,23)(21,26)(24,25);(4,5)(8,9)(13,18)(15,17)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7512476, C2×#7589116, C2×#7548479
maximal normal subgroup ofC2×#7512476, C2×#7589116, C2×#7548479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7414674);