C₂×#1393430 #72302571

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

Aliases

direct product

C₂×#1393430

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

32175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72280033, C₂×#72280420, C₂×#72279963
maximal normal subgroup of	C₂×#72280033, C₂×#72280420, C₂×#72279963

GAP

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

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

C2×#1393430 #72302571