#93714

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:

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

Polycyclic presentation:

2^9|a^2=d,c^2=gh,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cghi,d^b=deghi,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, 910);