C₂×#663637 #72339409

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

Aliases

direct product

C₂×#663637

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43573 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72257916, C₂×#72327602, C₂×#72261136, …
maximal normal subgroup of	C₂×#72257916, C₂×#72327602, C₂×#72261136, …

GAP

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

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

C2×#663637 #72339409