C₂×#7967693 #72298932

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

Aliases

direct product

C₂×#7967693

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

54693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297328
maximal normal subgroup of	C₂×#72297328
maximal quotient of	C₂×#72297592

GAP

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

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

C2×#7967693 #72298932