C₂×#873894 #72325524

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#873894

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42353 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)(13,18)(14,17)(15,16)(19,20,24,23)(21,26,22,25);(3,4,6,5)(8,10)(12,16)(13,17);(8,10)(19,21,24,22)(20,25,23,26);(7,8,9,10)(11,13)(12,14)(15,17)(16,18)(19,22,24,21)(20,26,23,25);(1,2);(3,5,6,4)(11,14)(12,17)(13,16)(15,18)(19,23,24,20)(21,26,22,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232179, C₂×#72232154, C₂×#72232149
maximal normal subgroup of	C₂×#72232179, C₂×#72232154, C₂×#72232149

GAP

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

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

C2×#873894 #72325524