#94218

order	512
nilpotency class	5
isoclinism	isoclinic class of #94210
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 167 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)(6,31)(7,34)(8,13)(9,16,36,54)(10,25,40,72)(11,43)(12,32)(14,35)(17,58)(18,62)(19,65)(20,51)(22,44)(24,74,71,41)(26,78,73,69)(28,83)(29,45)(30,81)(33,87)(37,92,90,126)(38,95,91,105)(39,96,88,57)(42,70,89,77)(46,84)(47,85)(48,56,100,107)(49,98)(50,63)(52,66)(53,101)(55,108,106,79)(59,114)(60,102)(61,112)(64,118)(68,121,120,99)(75,124,122,80)(76,97,123,119)(82,86)(93,110,125,128)(94,127,111,109)(103,115)(104,116)(113,117);(1,3,5,17,6,18,28,59,7,19,29,60,32,63,84,115,8,20,30,61,33,64,85,116,35,66,86,117,22,53,43,98)(2,9,11,37,44,91,82,125,14,39,47,94,87,100,81,108,13,36,46,90,12,38,45,93,34,88,83,111,31,48,27,79)(4,16,24,55,25,56,75,109,26,57,76,110,77,95,121,126,23,54,71,106,72,107,122,127,73,96,123,128,70,105,99,92)(10,15,41,49,67,101,120,113,42,52,97,104,78,118,124,112,40,51,74,103,21,50,68,102,89,65,119,114,69,62,80,58)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,e^2=fg,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, 1414);