#9910680

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3758 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

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, 9817876);