#9481520

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

3637 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)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32)(33,34,38,42)(35,43,46,39)(36,40,49,56)(37,50,51,41)(44,54,47,52)(45,53,59,57)(48,61,62,55)(58,63,60,64);(2,6)(7,9)(8,13)(14,15)(18,23)(19,25)(24,30)(26,31)(33,34,38,42)(35,39,46,43)(36,40,49,56)(37,41,51,50)(44,52,47,54)(45,53,59,57)(48,55,62,61)(58,63,60,64);(4,12)(8,13)(10,16)(14,15)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)(33,34)(35,43)(36,40)(37,50)(38,42)(39,46)(41,51)(44,54)(45,53)(47,52)(48,61)(49,56)(55,62)(57,59)(58,63)(60,64);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,35,38,46)(34,39,42,43)(36,47,49,44)(37,45,51,59)(40,54,56,52)(41,53,50,57)(48,60,62,58)(55,64,61,63);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(33,36,38,49)(34,40,42,56)(35,44,46,47)(37,48,51,62)(39,52,43,54)(41,55,50,61)(45,58,59,60)(53,63,57,64);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,37)(34,41)(35,45)(36,48)(38,51)(39,53)(40,55)(42,50)(43,57)(44,58)(46,59)(47,60)(49,62)(52,63)(54,64)(56,61)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,e^2=h,f^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, 9388716);