C₂×#581605 #72338033

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

Aliases

direct product

C₂×#581605

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

54293 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232342, C₂×#72210496, C₂×#72232292
maximal normal subgroup of	C₂×#72232342, C₂×#72210496, C₂×#72232292

GAP

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

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

C2×#581605 #72338033