C₂×#663658 #72337828

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

Aliases

direct product

C₂×#663658

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

50249 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)(13,17)(14,15)(16,18)(20,21)(24,25);(7,8,9,10)(12,15)(16,17)(19,20)(21,23)(22,25)(24,26);(3,4,5,6)(7,8)(9,10)(13,18)(16,17)(20,21)(22,26);(3,4)(5,6)(7,8)(9,10)(11,13,14,18)(12,16,15,17)(19,21,23,20)(22,24,26,25);(1,2);(11,12)(13,17)(14,15)(16,18)(19,22)(20,24)(21,25)(23,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232284, C₂×#72232322, C₂×#72232291
maximal normal subgroup of	C₂×#72232284, C₂×#72232322, C₂×#72232291

GAP

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

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

C2×#663658 #72337828