#9481448

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(19,25)(23,30)(26,28)(31,32)(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);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(34,37)(35,44)(41,47)(42,51)(45,48)(46,58)(56,60)(59,61);(5,13)(8,15)(10,16)(12,14)(18,22)(23,30)(24,27)(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);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(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);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(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);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(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=hi,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, 9388644);