C₂×#8272309 #72294558

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8272309

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 9794 conjugacy classes of subgroups; only the 29501 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72293940, C₂×#72293942, C₂×#72293944

GAP

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

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

C2×#8272309 #72294558