C₂×#7685071 #72279829

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7685071

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

116975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279669
maximal normal subgroup of	C₂×#72279669

GAP

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

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

C2×#7685071 #72279829