#8417210

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,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, 8324406);