C₂×#1259673 #72334722

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

Aliases

direct product

C₂×#1259673

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67643 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334367, C₂×#72334373, C₂×#72334378
maximal normal subgroup of	C₂×#72334367, C₂×#72334373, C₂×#72334378
maximal quotient of	C₂×#72334398

GAP

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

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

C2×#1259673 #72334722