#8417216

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

3709 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)(3,10)(4,7)(5,12)(6,9)(8,14)(11,13)(15,16)(17,18,23,19)(20,27,30,24)(21,25,22,26)(28,32,29,31)(35,43)(39,44)(45,48)(46,57)(53,55)(54,58)(59,61)(63,64);(2,4)(3,9)(8,12)(11,15)(18,19)(24,27)(25,26)(31,32)(33,34,38,42)(35,44,43,39)(36,47,50,40)(37,41,52,51)(45,53,48,55)(46,58,57,54)(49,60,62,56)(59,63,61,64);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,23,18)(20,27,30,24)(21,26,22,25)(28,32,29,31)(33,34)(35,39)(36,40)(37,51)(38,42)(41,52)(43,44)(45,53)(46,58)(47,50)(48,55)(49,60)(54,57)(56,62)(59,64)(61,63);(1,4,7,2)(3,10,9,6)(5,12,14,8)(11,16,15,13)(17,20,23,30)(18,24,19,27)(21,28,22,29)(25,31,26,32)(33,35,38,43)(34,39,42,44)(36,48,50,45)(37,46,52,57)(40,55,47,53)(41,54,51,58)(49,61,62,59)(56,64,60,63);(1,5,7,14)(2,8,4,12)(3,11,9,15)(6,13,10,16)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)(33,36,38,50)(34,40,42,47)(35,45,43,48)(37,49,52,62)(39,53,44,55)(41,56,51,60)(46,59,57,61)(54,63,58,64);(1,6,7,10)(2,9,4,3)(5,13,14,16)(8,15,12,11)(17,22,23,21)(18,26,19,25)(20,29,30,28)(24,32,27,31)(33,37)(34,41)(35,46)(36,49)(38,52)(39,54)(40,56)(42,51)(43,57)(44,58)(45,59)(47,60)(48,61)(50,62)(53,63)(55,64)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=i,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, 8324412);