#411587

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^c=fi

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, 318783);