C₂×#1086847 #72296795

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#1086847

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

52125 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278277
maximal normal subgroup of	C₂×#72278277
maximal quotient of	C₂×#72278761

GAP

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

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

C2×#1086847 #72296795