C₂×#8429708 #72331964

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

Aliases

direct product

C₂×#8429708

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 9648 conjugacy classes of subgroups; only the 29533 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)(9,10);(3,4)(5,10)(6,9)(7,8)(12,13)(14,18)(20,21)(22,26);(3,5,7,6)(4,9,8,10)(12,13)(16,17)(19,20,23,21)(22,24,26,25);(4,8)(5,6)(11,12,15,13)(14,17,18,16)(19,20,23,21)(22,25,26,24);(1,2);(3,6,7,5)(4,10,8,9)(11,13,15,12)(14,16,18,17)(19,21,23,20)(22,25,26,24);(3,4)(5,10)(6,9)(7,8)(11,14)(12,16)(13,17)(15,18)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232288
maximal normal subgroup of	C₂×#72232288
maximal quotient of	C₂×#72323410

GAP

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

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

C2×#8429708 #72331964