#9910663

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3737 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)(33,34,39,43)(35,44,40,36)(37,41,51,57)(38,52,53,42)(45,55,49,47)(46,48,54,58)(50,62,63,56)(59,60,61,64);(2,5)(8,9)(10,13)(15,16)(18,22)(23,29)(24,27)(30,32)(33,35,39,40)(34,36,43,44)(37,49,51,45)(38,46,53,54)(41,55,57,47)(42,48,52,58)(50,61,63,59)(56,64,62,60);(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)(33,35)(34,36)(37,45)(38,46)(39,40)(41,47)(42,48)(43,44)(49,51)(50,59)(52,58)(53,54)(55,57)(56,60)(61,63)(62,64);(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)(33,36)(34,40)(35,43)(37,47)(38,48)(39,44)(41,49)(42,54)(45,57)(46,52)(50,60)(51,55)(53,58)(56,61)(59,62)(63,64);(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)(33,37,39,51)(34,41,43,57)(35,45,40,49)(36,47,44,55)(38,50,53,63)(42,56,52,62)(46,59,54,61)(48,60,58,64);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38,39,53)(34,42,43,52)(35,46,40,54)(36,48,44,58)(37,50,51,63)(41,56,57,62)(45,59,49,61)(47,60,55,64)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,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^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, 9817859);