#93713

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₂×M₁₆	C₂×C₈
Frattini, cosocle	#2151	C₂²
socle	C₂	#7060
prime radical	#575	C₈
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2151 ◁ G
LECS	G ▷ #2151 ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

128 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=ghi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgh,d^b=degh,g^b=gi,d^c=dg,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 909);