C₂×#636791 #72324452

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#636791

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

83565 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)(13,14)(15,16)(17,18)(20,21)(24,25);(4,6)(8,10)(19,20,23,21)(22,25,26,24);(12,16)(14,17);(4,6)(8,10)(11,13)(12,17)(14,16)(15,18)(19,21,23,20)(22,25,26,24);(1,2);(3,4,5,6)(7,8)(9,10)(11,14,15,17)(12,13,16,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72225665, C₂×#72232045, C₂×#72232042, …
maximal normal subgroup of	C₂×#72225665, C₂×#72232045, C₂×#72232042, …
maximal quotient of	C₂×#72298421, #72335067, C₂×#72323618

GAP

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

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

C2×#636791 #72324452