C₂×#7633151 #72339953

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#7633151

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

306629 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)(8,10)(12,17)(15,18)(19,25)(22,24);(11,12,16,17)(13,18,20,15)(14,19,23,25)(21,22,26,24);(4,6)(8,10)(11,13,16,20)(12,18,17,15)(14,21,23,26)(19,22,25,24);(8,10)(11,13)(12,18)(14,21)(15,17)(16,20)(19,22)(23,26)(24,25);(8,10)(11,14)(12,19)(13,21)(15,24)(16,23)(17,25)(18,22)(20,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,15,16,18)(12,20,17,13)(14,22,23,24)(19,21,25,26)

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72337230, #72339970, C₂×#72337324

GAP

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

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

C2×#7633151 #72339953