#9911007

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3739 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:

(1,2,6,7)(3,4,8,11)(5,9,14,15)(10,12,16,13)(19,25)(23,29)(26,31)(30,32)(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);(2,7)(4,11)(9,15)(12,13)(18,22)(23,29)(24,27)(30,32)(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);(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(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);(1,3,6,8)(2,4,7,11)(5,10,14,16)(9,12,15,13)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(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);(1,4)(2,8)(3,7)(5,13)(6,11)(9,10)(12,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(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);(1,5,6,14)(2,9,7,15)(3,10,8,16)(4,12,11,13)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(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=h,d^2=g,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, 9818203);