C₂×#1572545 #72325226

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#1572545

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

47361 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232169, C₂×#72232110, C₂×#72232134
maximal normal subgroup of	C₂×#72232169, C₂×#72232110, C₂×#72232134

GAP

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

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

C2×#1572545 #72325226