C₂×#9715478 #72298978

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

Aliases

direct product

C₂×#9715478

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

58031 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:

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

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297217, C₂×#72232288
maximal normal subgroup of	C₂×#72297217, C₂×#72232288
maximal quotient of	C₂×#72297588, C₂×#72323417

GAP

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

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

C2×#9715478 #72298978