C₂×#8093625 #72294071

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

Aliases

direct product

C₂×#8093625

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

73909 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241247
maximal normal subgroup of	C₂×#72241247
maximal quotient of	C₂×#72293921, C₂×#72293924, C₂×#72293944, …

GAP

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

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

C2×#8093625 #72294071