C₂×#800398 #72302487

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#800398

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

33079 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72279950, C₂×#72279836, C₂×#72280397, …
maximal normal subgroup of	C₂×#72279950, C₂×#72279836, C₂×#72280397, …

GAP

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

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

C2×#800398 #72302487