C₂×#5955307 #72324403

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂×#5955307

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

61777 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232053, C₂×#72232045, C₂×#72232056
maximal normal subgroup of	C₂×#72232053, C₂×#72232045, C₂×#72232056

GAP

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

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

C2×#5955307 #72324403