C₂×#8438555 #72331969

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

Aliases

direct product

C₂×#8438555

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 6456 conjugacy classes of subgroups; only the 29613 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323406

GAP

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

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

C2×#8438555 #72331969