C₂×#2110419 #72296652

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#2110419

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

43439 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)(13,17)(15,18)(19,20)(21,22)(23,24)(25,26);(3,4,5,6)(7,8,9,10)(11,12,14,16)(13,15,17,18)(20,24)(22,26);(4,6)(7,8)(9,10)(13,17)(15,18)(19,21,23,25)(20,22,24,26);(3,4)(5,6)(7,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,21)(20,22)(23,25)(24,26);(11,13)(12,15)(14,17)(16,18)(19,22)(20,25)(21,24)(23,26);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278611, C₂×#72232122, C₂×#72323252, …
maximal normal subgroup of	C₂×#72278611, C₂×#72232122, C₂×#72323252, …
maximal quotient of	C₂×#72278779, C₂×#72278668, C₂×#72278082

GAP

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

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

C2×#2110419 #72296652