#9481451

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

3648 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388647);