C₂×#846465 #72323851

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#846465

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

72161 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323251, C₂×#72232176, C₂×#72237111, …
maximal normal subgroup of	C₂×#72323251, C₂×#72232176, C₂×#72237111, …
maximal quotient of	C₂×#72332588, C₂×#72298452, C₂×#72298450

GAP

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

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

C2×#846465 #72323851