#1350×C₂₁₁ #41008970

order	27008
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₂₁₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

#1350×C₂₁₁

Subgroups

	subgroup	G/H
center, socle	C₂²×C₂₁₁	#102
commutator	C₂×C₈	C₂×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₈	C₂²×C₂₁₁
prime radical	C₂³×C₂₁₁	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ C₂²×C₈×C₂₁₁ ◁ G

104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 251
transitive degree	μ_t(G) = 13504

A minimal faithful permutation representation — 2 generators on 251 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,18,15,20,14,19)(11,21,23,34,25,38,24,37)(12,26,28,17,30,35,29,36)(16,27,31,22,33,39,32,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,39)(13,24,14,23)(17,34,36,37)(18,32,19,31)(21,26,38,35)(22,28,40,29)

Conjugacy classes & character table

6119 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27008, 82);