#8417044

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

3707 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,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, 8324240);