#8417012

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

3709 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,39,43)(35,40,44,37)(36,47,48,41)(38,42,54,58)(45,51,56,49)(46,55,53,52)(50,60,61,57)(59,64,63,62);(2,3)(7,9)(8,10)(14,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(35,44)(36,48)(37,40)(41,47)(46,53)(50,61)(52,55)(57,60);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,40)(36,45)(37,43)(38,53)(39,44)(41,49)(42,52)(46,54)(47,51)(48,56)(50,63)(55,58)(57,64)(59,61)(60,62);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,39,48)(34,41,43,47)(35,45,44,56)(37,51,40,49)(38,50,54,61)(42,57,58,60)(46,59,53,63)(52,64,55,62);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(33,37,39,40)(34,35,43,44)(36,49,48,51)(38,52,54,55)(41,56,47,45)(42,46,58,53)(50,62,61,64)(57,63,60,59);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38)(34,42)(35,46)(36,50)(37,52)(39,54)(40,55)(41,57)(43,58)(44,53)(45,59)(47,60)(48,61)(49,62)(51,64)(56,63)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324208);