C₂×#641347 #72324001

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄³

Aliases

direct product

C₂×#641347

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

51429 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297237, C₂×#72297691, C₂×#72323264
maximal normal subgroup of	C₂×#72297237, C₂×#72297691, C₂×#72323264

GAP

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

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

C2×#641347 #72324001