C₂×#4308620 #72334688

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

Aliases

direct product

C₂×#4308620

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

68803 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)(7,8,10,9)(13,16)(14,18)(19,20,22,24)(21,23,25,26);(4,5)(8,9)(11,12,15,17)(13,18,16,14)(21,25)(23,26);(3,4,6,5)(7,8,10,9)(13,16)(14,18)(19,20)(21,26)(22,24)(23,25);(3,5,6,4)(7,9,10,8)(11,13)(12,14)(15,16)(17,18)(19,21,22,25)(20,23,24,26);(1,2);(11,14)(12,16)(13,17)(15,18)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334344, C₂×#72334345, C₂×#72309280
maximal normal subgroup of	C₂×#72334344, C₂×#72334345, C₂×#72309280
maximal quotient of	C₂×#72334387, C₂×#72334409, C₂×#72334413

GAP

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

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

C2×#4308620 #72334688