C₂×#576038 #72338025

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

Aliases

direct product

C₂×#576038

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

55585 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)(7,8,9,10)(12,15)(13,17)(19,20,23,24)(21,25,26,22);(11,12)(13,18)(14,15)(16,17)(19,20,23,24)(21,22,26,25);(3,4,6,5)(7,8)(9,10)(12,15)(16,18)(20,24)(21,26);(1,2);(8,10)(11,12)(13,18)(14,15)(16,17)(19,21)(20,22)(23,26)(24,25);(3,5,6,4)(7,8)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22)(20,21)(23,25)(24,26)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210448, C₂×#72232354, C₂×#72232292
maximal normal subgroup of	C₂×#72210448, C₂×#72232354, C₂×#72232292

GAP

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

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

C2×#576038 #72338025