C₂×#1393283 #72325534

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#1393283

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

42169 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232179, C₂×#72232123, C₂×#72232119
maximal normal subgroup of	C₂×#72232179, C₂×#72232123, C₂×#72232119

GAP

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

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

C2×#1393283 #72325534