C₂×#1506849 #72294273

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#1506849

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

50917 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293907, C₂×#72293942, C₂×#72232681, …
maximal normal subgroup of	C₂×#72293907, C₂×#72293942, C₂×#72232681, …
maximal quotient of	C₂×#72293932, C₂×#72293982

GAP

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

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

C2×#1506849 #72294273