#94605

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	D₈
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle	C₂	D₄⋊C₃₂
prime radical	#135	C₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,5,12,7,14,31,49,8,11,32,44,35,45,80,94,6,15,30,50,34,55,79,101,29,46,77,95,78,97,117,124,9,16,33,51,36,56,81,102,37,47,82,96,83,98,118,125,26,13,52,48,75,54,103,100,4,24,27,71,28,72,76,115)(3,17,19,57,21,58,64,105,18,10,38,39,60,41,84,88,22,42,65,89,69,92,112,122,61,40,85,87,108,91,119,121,23,59,66,106,70,107,113,127,62,43,86,90,109,93,120,123,20,67,63,110,68,114,111,128,25,53,73,99,74,104,116,126);(1,3)(2,10)(4,25)(5,19)(6,22)(7,21)(8,18)(9,23)(11,42)(12,39)(13,53)(14,41)(15,40)(16,43)(17,24)(20,26)(27,73)(28,74)(29,61)(30,65)(31,64)(32,38)(33,66)(34,69)(35,60)(36,70)(37,62)(44,89)(45,92)(46,59)(47,67)(48,99)(49,88)(50,87)(51,90)(52,63)(54,104)(55,91)(56,93)(57,71)(58,72)(68,75)(76,116)(77,85)(78,108)(79,112)(80,84)(81,113)(82,86)(83,109)(94,122)(95,106)(96,110)(97,107)(98,114)(100,126)(101,121)(102,123)(103,111)(105,115)(117,119)(118,120)(124,127)(125,128)

Polycyclic presentation:

2^9|a^2=d,c^2=ei,d^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=ce,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1801);