#94211

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1384
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	#2108	C₂²
socle	C₂	#7218
prime radical	C₂³	#359
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄² ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ #2108 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

108 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fghi,e^2=fhi,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, 1407);