C₂×#1506926 #72325165

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

Aliases

direct product

C₂×#1506926

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

42177 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232089, C₂×#72232116, C₂×#72232164
maximal normal subgroup of	C₂×#72232089, C₂×#72232116, C₂×#72232164

GAP

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

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

C2×#1506926 #72325165