#147689

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,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, 54885);