C₂×#1393405 #72302570

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

Aliases

direct product

C₂×#1393405

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

27095 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)(7,8,10,9)(20,22)(21,23);(3,4,5,6)(13,17)(16,18)(19,20,24,22)(21,26,23,25);(7,9)(8,10)(11,12,15,14)(13,16,17,18)(19,21,24,23)(20,25,22,26);(3,4)(5,6)(7,9,10,8)(21,23)(25,26);(3,4)(5,6)(7,9)(8,10)(11,13)(12,16)(14,18)(15,17)(19,22,24,20)(21,26,23,25);(3,4)(5,6)(7,9)(8,10)(11,14,15,12)(13,18,17,16)(19,23,24,21)(20,25,22,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279903, C₂×#72280420, C₂×#72279926
maximal normal subgroup of	C₂×#72279903, C₂×#72280420, C₂×#72279926

GAP

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

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

C2×#1393405 #72302570