C₂×#1006126 #72325448

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

Aliases

direct product

C₂×#1006126

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

53123 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232097, C₂×#72232155, C₂×#72232119, …
maximal normal subgroup of	C₂×#72232097, C₂×#72232155, C₂×#72232119, …

GAP

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

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

C2×#1006126 #72325448