C₂×#1420730 #72337869

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

Aliases

direct product

C₂×#1420730

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

55065 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210448, C₂×#72232339, C₂×#72232330
maximal normal subgroup of	C₂×#72210448, C₂×#72232339, C₂×#72232330

GAP

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

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

C2×#1420730 #72337869