#8417036

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

3713 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)(3,5,10,8)(4,12,14,9)(11,15,16,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31)(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);(2,7)(3,10)(9,12)(11,16)(17,18,22,26)(19,23,30,27)(20,24,21,25)(28,31,29,32)(35,44)(36,48)(37,40)(41,47)(46,53)(50,61)(52,55)(57,60);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(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);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27)(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);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,27,32)(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);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21,22,20)(18,25,26,24)(19,29,30,28)(23,32,27,31)(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=ghi,b^2=gh,c^2=h,d^2=gi,e^2=i,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, 8324232);