#147678

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

997 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

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

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, 54874);