#9390428

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3744 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)(3,11)(4,15)(5,12)(6,16)(7,10)(8,14)(9,13)(17,18,23,20)(19,26,25,22)(21,28,30,24)(27,29,31,32)(33,34,39,40)(35,36,44,41)(37,42,49,53)(38,51,52,43)(45,47,57,54)(46,55,58,48)(50,62,61,56)(59,63,64,60);(1,3)(2,8)(4,12)(5,6)(7,14)(9,16)(10,11)(13,15)(17,18,23,20)(19,26,25,22)(21,28,30,24)(27,29,31,32)(34,40)(35,44)(37,49)(43,51)(46,58)(47,54)(50,61)(60,63);(1,4)(2,6)(3,12)(5,8)(7,16)(9,14)(10,15)(11,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,39,44)(34,41,40,36)(37,45,49,57)(38,46,52,58)(42,54,53,47)(43,55,51,48)(50,59,61,64)(56,63,62,60);(1,5,7,13)(2,9,10,12)(3,6,14,15)(4,8,16,11)(17,20,23,18)(19,26,25,22)(21,24,30,28)(27,29,31,32)(33,36,39,41)(34,35,40,44)(37,47,49,54)(38,48,52,55)(42,45,53,57)(43,46,51,58)(50,60,61,63)(56,59,62,64);(1,4)(2,6)(3,9)(5,11)(7,16)(8,13)(10,15)(12,14)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37)(34,42)(35,45)(36,47)(38,50)(39,49)(40,53)(41,54)(43,56)(44,57)(46,59)(48,60)(51,62)(52,61)(55,63)(58,64);(1,6)(2,4)(3,13)(5,14)(7,15)(8,9)(10,16)(11,12)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38)(34,43)(35,46)(36,48)(37,50)(39,52)(40,51)(41,55)(42,56)(44,58)(45,59)(47,60)(49,61)(53,62)(54,63)(57,64)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297624);