C₂×#779788 #72325386

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#779788

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

40529 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)(11,12,16,15)(13,17,18,14)(19,20)(21,25)(22,23)(24,26);(3,4,5,6)(7,8)(9,10)(11,13,16,18)(12,17,15,14)(19,21)(20,24)(22,25)(23,26);(4,6)(11,14,16,17)(12,13,15,18)(19,21)(20,25)(22,24)(23,26);(7,8,9,10)(11,14)(12,13)(15,18)(16,17)(19,22,23,20)(21,25,26,24);(1,2);(3,4)(5,6)(7,8)(9,10)(11,15,16,12)(13,17,18,14)(19,21)(20,25)(22,24)(23,26)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232179, C₂×#72232131, C₂×#72232109
maximal normal subgroup of	C₂×#72232179, C₂×#72232131, C₂×#72232109

GAP

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

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

C2×#779788 #72325386