C₂×#5955343 #72339417

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#5955343

Subgroups

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

52005 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327586, C₂×#72327582, C₂×#72327623
maximal normal subgroup of	C₂×#72327586, C₂×#72327582, C₂×#72327623

GAP

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

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

C2×#5955343 #72339417