C₂×#7837054 #72304802

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

Aliases

direct product

C₂×#7837054

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 9412 conjugacy classes of subgroups; only the 29733 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232290
maximal normal subgroup of	C₂×#72232290

GAP

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

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

C2×#7837054 #72304802