C₂×#10586422 #72331658

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂×#10586422

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁷
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

481285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₂×#72327577, C₂×#72241227, C₂×#72241247

GAP

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

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

C2×#10586422 #72331658