C₂×#8934366 #72278255

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#8934366

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

85221 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72237717, C₂×#72240900, C₂×#72240899
maximal subgroup of	#72278867, C₂²×#8934366, C₂×#72236189, …
maximal normal subgroup of	#72278867, C₂²×#8934366, C₂×#72236189, …
maximal quotient of	C₂×#72231761, C₂×#72230894, C₂×#72232875, …

GAP

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

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

C2×#8934366 #72278255