C₂×#8689724 #72332002

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂×#8689724

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

56979 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323417

GAP

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

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

C2×#8689724 #72332002