C₂×#7850986 #72329338

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

Aliases

direct product

C₂×#7850986

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 8848 conjugacy classes of subgroups; only the 29725 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:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232285
maximal normal subgroup of	C₂×#72232285

GAP

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

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

C2×#7850986 #72329338