C₂×#774690 #72302525

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

Aliases

direct product

C₂×#774690

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

30347 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279803, C₂×#72280075, C₂×#72280180
maximal normal subgroup of	C₂×#72279803, C₂×#72280075, C₂×#72280180

GAP

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

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

C2×#774690 #72302525