C₂×#574893 #72334635

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#574893

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71129 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233713, C₂×#72233708, C₂×#72233704, …
maximal normal subgroup of	C₂×#72233713, C₂×#72233708, C₂×#72233704, …
maximal quotient of	C₂×#72252090, #72337356

GAP

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

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

C2×#574893 #72334635