#8417054

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324250);