C₂×#1393187 #72325536

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

Aliases

direct product

C₂×#1393187

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

55525 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232181, C₂×#72232123, C₂×#72232117
maximal normal subgroup of	C₂×#72232181, C₂×#72232123, C₂×#72232117

GAP

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

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

C2×#1393187 #72325536