C₂×#2735950 #72294780

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

Aliases

direct product

C₂×#2735950

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

49165 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293946, C₂×#72232087, C₂×#72232123, …
maximal normal subgroup of	C₂×#72293946, C₂×#72232087, C₂×#72232123, …
maximal quotient of	C₂×#72293970

GAP

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

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

C2×#2735950 #72294780