C₂×#1258906 #72335178

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1258906

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67611 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72335121, C₂×#72335122, C₂×#72335124
maximal normal subgroup of	C₂×#72335121, C₂×#72335122, C₂×#72335124
maximal quotient of	C₂×#72335143

GAP

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

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

C2×#1258906 #72335178