C₂×#7630463 #72330538

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#7630463

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

157629 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72267577
maximal normal subgroup of	C₂×#72267577
maximal quotient of	C₂×#72323283

GAP

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

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

C2×#7630463 #72330538