C₂×#2606481 #72325540

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

Aliases

direct product

C₂×#2606481

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

45429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232138, C₂×#72232146, C₂×#72232182
maximal normal subgroup of	C₂×#72232138, C₂×#72232146, C₂×#72232182

GAP

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

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

C2×#2606481 #72325540