#147965

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55161);