C₂×#7687342 #72298907

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

Aliases

direct product

C₂×#7687342

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂×#35680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

58879 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:

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

Conjugacy classes & character table

226 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297796
maximal normal subgroup of	C₂×#72297796
maximal quotient of	C₂×#72298087

GAP

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

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

C2×#7687342 #72298907