C₂×#1426242 #72299217

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#1426242

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

43085 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232919, C₂×#72232497, C₂×#72297597
maximal normal subgroup of	C₂×#72232919, C₂×#72232497, C₂×#72297597

GAP

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

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

C2×#1426242 #72299217