C₂×#775617 #72325189

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

Aliases

direct product

C₂×#775617

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

49863 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)(7,8)(9,10)(14,18)(16,17)(19,20,24,23)(21,26,25,22);(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,17)(16,18)(19,21)(20,22)(23,26)(24,25);(4,5)(12,13)(14,18)(19,22,24,26)(20,25,23,21);(7,8,9,10)(11,13,15,12)(14,17,18,16)(19,22)(20,21)(23,25)(24,26);(1,2);(3,5,6,4)(7,8)(9,10)(11,14,15,18)(12,16,13,17)(19,23,24,20)(21,26,25,22)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232168, C₂×#72232135, C₂×#72232117
maximal normal subgroup of	C₂×#72232168, C₂×#72232135, C₂×#72232117

GAP

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

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

C2×#775617 #72325189