#147697

order	512
nilpotency class	4
isoclinism	isoclinic class of #137665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1895
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

1003 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,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, 54893);