#147701

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
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

1001 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)(3,14,18,44)(4,19,23,37)(5,25)(6,11)(7,13)(8,16,34,51)(9,22,38,24)(10,27)(12,30)(15,45)(17,39,29,43)(20,54)(21,56,59,36)(26,32,50,33)(28,40)(31,55)(35,46)(41,49)(42,47)(48,62)(52,64)(53,58,63,60)(57,61);(1,3)(2,8)(4,20)(5,26)(6,29)(7,18)(9,35)(10,39)(11,33)(12,32)(13,34)(14,40)(15,22)(16,49)(17,30)(19,52)(21,57)(23,47)(24,48)(25,44)(27,51)(28,50)(31,37)(36,54)(38,61)(41,43)(42,56)(45,53)(46,59)(55,58)(60,64)(62,63);(1,4)(2,9)(3,15)(5,21)(6,24)(7,23)(8,31)(10,36)(11,37)(12,19)(13,38)(14,42)(16,46)(17,47)(18,48)(20,29)(22,30)(25,53)(26,55)(27,60)(28,59)(32,35)(33,61)(34,52)(39,62)(40,63)(41,56)(43,45)(44,54)(49,58)(50,64)(51,57)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 54897);