C₂×#5020720 #72294170

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#5020720

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

130927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293904, C₂×#72277898
maximal normal subgroup of	C₂×#72293904, C₂×#72277898
maximal quotient of	C₂×#72278116

GAP

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

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

C2×#5020720 #72294170