#8417233

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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, 8324429);