C₂×#3858126 #72325305

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#3858126

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

40029 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232087, C₂×#72232158, C₂×#72232128
maximal normal subgroup of	C₂×#72232087, C₂×#72232158, C₂×#72232128

GAP

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

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

C2×#3858126 #72325305