C₂×#1394155 #72324047

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#1394155

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

73179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237111, C₂×#72297911, C₂×#72261233, …
maximal normal subgroup of	C₂×#72237111, C₂×#72297911, C₂×#72261233, …
maximal quotient of	C₂×#72298133, C₂×#72298373

GAP

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

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

C2×#1394155 #72324047