C₂×#7820289 #72331874

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

Aliases

direct product

C₂×#7820289

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 9872 conjugacy classes of subgroups; only the 29797 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323387

GAP

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

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

C2×#7820289 #72331874