C₂×#4929128 #72325281

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

Aliases

direct product

C₂×#4929128

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

56355 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232187, C₂×#72232113, C₂×#72232109
maximal normal subgroup of	C₂×#72232187, C₂×#72232113, C₂×#72232109

GAP

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

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

C2×#4929128 #72325281