C₂×#1345987 #72332671

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

Aliases

direct product

C₂×#1345987

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

37955 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)(14,18)(16,17)(19,20,23,25)(21,24,26,22);(7,8)(9,10)(11,12)(13,15)(14,16)(17,18);(3,4)(5,6)(12,13)(16,17)(19,21,23,26)(20,24,25,22);(1,2);(3,4,5,6)(11,13,15,12)(14,17,18,16)(19,22,23,24)(20,21,25,26);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,21)(20,24)(22,25)(23,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323393, C₂×#72323408, C₂×#72232349
maximal normal subgroup of	C₂×#72323393, C₂×#72323408, C₂×#72232349

GAP

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

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

C2×#1345987 #72332671