#94219

order	512
nilpotency class	5
isoclinism	isoclinic class of #94210
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1384
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×#504	C₂²
socle	C₂	#7230
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 128
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,e^2=fgi,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,d^b=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1415);