C₂×#7671478 #72302264

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

Aliases

direct product

C₂×#7671478

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

123833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268516
maximal normal subgroup of	C₂×#72268516
maximal quotient of	C₂×#72280318, C₂×#72279926, C₂×#72280221, …

GAP

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

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

C2×#7671478 #72302264