C₂×#1086877 #72324409

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#1086877

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

60347 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)(13,16)(14,15)(17,18)(21,25)(24,26);(13,17)(16,18)(19,20)(21,24)(22,23)(25,26);(4,6)(8,10)(12,15)(16,18)(20,23)(24,26);(8,10)(11,13)(12,16)(14,17)(15,18)(19,21,22,25)(20,24,23,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232044, C₂×#72232053, C₂×#72232047
maximal normal subgroup of	C₂×#72232044, C₂×#72232053, C₂×#72232047

GAP

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

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

C2×#1086877 #72324409