C₂×#2306447 #72297456

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

Aliases

direct product

C₂×#2306447

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

41613 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277929
maximal normal subgroup of	C₂×#72277929
maximal quotient of	C₂×#72278034

GAP

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

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

C2×#2306447 #72297456