#9390429

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3764 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 9297625);