#153391

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

162 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,18,28,43,57,40,13,19,29,44,32,48,59,42,15,21,30,45,60,70,61,46,31,47,58,41,14,20,27,17)(11,16,23,34,50,64,51,33,22,37,49,66,56,69,71,62,26,38,54,67,55,68,72,63,24,35,52,65,25,36,53,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,23)(13,22)(14,25)(15,26)(17,39)(18,34)(19,37)(20,36)(21,38)(24,31)(27,53)(28,50)(29,49)(30,54)(32,56)(33,40)(35,47)(41,65)(42,62)(43,64)(44,66)(45,67)(46,63)(48,69)(51,57)(52,58)(55,60)(59,71)(61,72)(68,70);(1,4,5,3)(2,6,7,8)(10,17)(11,22)(12,27)(13,31)(14,28)(16,33)(18,20)(19,46)(21,42)(23,51)(24,26)(25,56)(29,61)(30,59)(32,60)(34,64)(35,62)(36,66)(37,39)(38,63)(40,47)(41,43)(44,70)(45,48)(49,53)(52,71)(54,72)(57,58)(65,69)(67,68)

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=d,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,f^b=fi,e^c=eg,f^c=fi,g^c=gh,h^c=hi

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, 60587);