C₂×#786588 #72302691

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#786588

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280075, C₂×#72279836, C₂×#72280221
maximal normal subgroup of	C₂×#72280075, C₂×#72279836, C₂×#72280221

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×#786588 #72302691