C₂×#8346549 #72337800

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

Aliases

direct product

C₂×#8346549

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 9788 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232298

GAP

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

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

C2×#8346549 #72337800