C₂×#4929121 #72325063

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

Aliases

direct product

C₂×#4929121

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

65833 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232190, C₂×#72232113, C₂×#72232117
maximal normal subgroup of	C₂×#72232190, C₂×#72232113, C₂×#72232117

GAP

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

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

C2×#4929121 #72325063