#8114395

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8021591);