C₂×#773933 #72324017

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#773933

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41859 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)(13,15)(14,17)(16,18)(19,20,24,25)(21,23,22,26);(8,10)(12,13)(16,17)(20,25)(21,22);(7,8)(9,10)(19,21,24,22)(20,26,25,23);(3,4,6,5)(11,13,15,12)(14,16,18,17)(19,22,24,21)(20,26,25,23);(1,2);(3,5,6,4)(11,14,15,18)(12,16,13,17)(19,23,24,26)(20,22,25,21)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297911, C₂×#72323270, C₂×#72323258
maximal normal subgroup of	C₂×#72297911, C₂×#72323270, C₂×#72323258

GAP

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

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

C2×#773933 #72324017