C₂×#777400 #72325191

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

Aliases

direct product

C₂×#777400

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

49751 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)(20,23)(24,25);(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,24)(22,23)(25,26);(4,5)(11,13,15,17)(12,16,18,14)(19,21)(20,24)(22,26)(23,25);(7,8,9,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,25)(22,23)(24,26);(1,2);(3,5,6,4)(7,8)(9,10)(11,14,15,16)(12,17,18,13)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232172, C₂×#72232131, C₂×#72232117
maximal normal subgroup of	C₂×#72232172, C₂×#72232131, C₂×#72232117

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

C2×#777400 #72325191