C₂×#875301 #72325151

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

Aliases

direct product

C₂×#875301

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

55461 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232088, C₂×#72232173, C₂×#72232097, …
maximal normal subgroup of	C₂×#72232088, C₂×#72232173, C₂×#72232097, …

GAP

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

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

C2×#875301 #72325151