C₂×#8688877 #72324535

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

Aliases

direct product

C₂×#8688877

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

72181 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:

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

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232069, C₂×#72232288
maximal normal subgroup of	C₂×#72232069, C₂×#72232288
maximal quotient of	C₂×#72232154, C₂×#72232157, C₂×#72323420

GAP

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

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

C2×#8688877 #72324535