C₂×#7684310 #72279802

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

Aliases

direct product

C₂×#7684310

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

115001 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:

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

C2×#7684310 #72279802