C₂×#8160007 #72331828

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

Aliases

direct product

C₂×#8160007

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 9766 conjugacy classes of subgroups; only the 29525 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323396

GAP

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

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

C2×#8160007 #72331828