C₂×#7685396 #72323844

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

Aliases

direct product

C₂×#7685396

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 6262 conjugacy classes of subgroups; only the 30149 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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232276
maximal normal subgroup of	C₂×#72232276
maximal quotient of	C₂×#72323246, C₂×#72323254, C₂×#72323258, …

GAP

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

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

C2×#7685396 #72323844