#94127

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×#504	C₂²
socle	C₂	#7199
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

167 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fi,d^b=dfi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1323);