C₂×#689016 #72325113

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

Aliases

direct product

C₂×#689016

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

49145 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232190, C₂×#72232131, C₂×#72232134
maximal normal subgroup of	C₂×#72232190, C₂×#72232131, C₂×#72232134

GAP

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

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

C2×#689016 #72325113