C₂×#8139517 #72337770

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8139517

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

At least 10288 conjugacy classes of subgroups; only the 29541 normal subgroups are shown.

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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232275
maximal normal subgroup of	C₂×#72232275
maximal quotient of	C₂×#72232311

GAP

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

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

C2×#8139517 #72337770