C₂×#154177 #72334748

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#154177

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂⁴	C₂⁴×C₄
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

69381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72250280, C₂×#72250292, C₂×#72334409, …
maximal normal subgroup of	C₂×#72250280, C₂×#72250292, C₂×#72334409, …
maximal quotient of	#72337397

GAP

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

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

C2×#154177 #72334748