#8926600

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833796);