C₂×#647924 #72325634

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#647924

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40673 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)(11,12,15,14)(13,16,17,18)(19,20,23,21)(22,24,26,25);(8,9)(11,13)(12,16)(14,18)(15,17)(20,21)(22,26);(4,6)(8,9)(19,20,23,21)(22,25,26,24);(1,2);(7,9,10,8)(11,13)(12,16)(14,18)(15,17)(19,21,23,20)(22,24,26,25);(3,4)(5,6)(11,14,15,12)(13,16,17,18)(19,22,23,26)(20,24,21,25)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232164, C₂×#72232161, C₂×#72232186
maximal normal subgroup of	C₂×#72232164, C₂×#72232161, C₂×#72232186

GAP

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

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

C2×#647924 #72325634