C₂×#7645258 #72331707

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

Aliases

direct product

C₂×#7645258

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

68977 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)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,25)(23,26);(4,7)(8,10)(12,16)(13,17)(19,21)(20,23)(22,26)(24,25);(19,20)(21,24)(22,25)(23,26);(1,2);(19,20)(21,23)(22,25)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,14,16,18)(19,21)(20,24)(22,26)(23,25);(3,5,6,9)(4,8,7,10)(11,14)(12,17)(13,16)(15,18)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227, C₂×#72241247
maximal normal subgroup of	C₂×#72241227, C₂×#72241247
maximal quotient of	C₂×#72327626

GAP

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

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

C2×#7645258 #72331707