C₂×#7648677 #72331679

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

Aliases

direct product

C₂×#7648677

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

73149 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72327586

GAP

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

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

C2×#7648677 #72331679