C₂×#1087741 #72337919

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

Aliases

direct product

C₂×#1087741

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

45331 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232339, C₂×#72232311, C₂×#72232342
maximal normal subgroup of	C₂×#72232339, C₂×#72232311, C₂×#72232342

GAP

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

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

C2×#1087741 #72337919