C₂×#4034147 #72337959

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

Aliases

direct product

C₂×#4034147

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

41453 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)(7,8,10,9)(11,12,14,15)(13,16,17,18)(19,20,22,24)(21,25,26,23);(4,5)(7,8,10,9)(12,15)(13,17)(19,20,22,24)(21,23,26,25);(4,5)(8,9)(21,26)(23,25);(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,26);(1,2);(3,5,6,4)(7,9,10,8)(11,13)(12,16)(14,17)(15,18)(19,21,22,26)(20,23,24,25)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232296, C₂×#72232298
maximal normal subgroup of	C₂×#72232281, C₂×#72232296, C₂×#72232298

GAP

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

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

C2×#4034147 #72337959