C₂×#7642250 #72342131

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

Aliases

direct product

C₂×#7642250

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

71665 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72342077
maximal normal subgroup of	C₂×#72342077

GAP

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

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

C2×#7642250 #72342131