#153392

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.

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

Polycyclic presentation:

2^9|a^2=ei,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, 60588);