#9911003

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3744 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)(7,13)(10,15)(14,16)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,32,29)(33,34)(35,40)(36,46)(37,41)(38,51)(39,43)(42,52)(44,53)(45,59)(47,55)(48,54)(49,61)(50,57)(56,62)(58,64)(60,63);(2,6)(7,13)(8,11)(14,16)(18,23)(20,27)(25,31)(28,29)(33,35,39,36)(34,40,43,46)(37,47,50,44)(38,45,52,48)(41,55,57,53)(42,54,51,59)(49,60,62,58)(56,64,61,63);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,24)(18,20,23,27)(21,26,30,32)(25,28,31,29)(33,36,39,35)(34,40,43,46)(37,47,50,44)(38,48,52,45)(41,53,57,55)(42,54,51,59)(49,60,62,58)(56,63,61,64);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,53)(45,54)(47,55)(48,59)(49,56)(50,57)(51,52)(58,63)(60,64)(61,62);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20)(18,24)(19,23)(21,29)(22,27)(25,26)(28,30)(31,32)(33,37,39,50)(34,41,43,57)(35,44,36,47)(38,49,52,62)(40,53,46,55)(42,56,51,61)(45,58,48,60)(54,63,59,64);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,25,23,31)(19,26,24,32)(20,28,27,29)(33,38)(34,42)(35,45)(36,48)(37,49)(39,52)(40,54)(41,56)(43,51)(44,58)(46,59)(47,60)(50,62)(53,63)(55,64)(57,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,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^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, 9818199);