#9481407

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

3643 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gi,e^2=g,f^2=h,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, 9388603);