C₂×#712420 #72302847

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#712420

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

39855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280535, C₂×#72280339, C₂×#72280221
maximal normal subgroup of	C₂×#72280535, C₂×#72280339, 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)(8,10)(12,16)(14,18)(19,20,23,21)(22,24,26,25), (7,8,9,10)(22,26)(24,25), (4,6)(8,10)(12,16)(14,18)(19,21,23,20)(22,25,26,24), (11,12,15,16)(13,14,17,18)(19,22)(20,24)(21,25)(23,26), (11,13)(12,14)(15,17)(16,18), (3,4)(5,6)(7,8)(9,10)(11,14)(12,17)(13,16)(15,18)(19,21)(20,23)(22,25)(24,26));

C2×#712420 #72302847