C₂×#663902 #72338036

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#663902

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

50927 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232322, C₂×#72232291, C₂×#72232346
maximal normal subgroup of	C₂×#72232322, C₂×#72232291, C₂×#72232346

GAP

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

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

C2×#663902 #72338036