#8115434

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

3735 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)(3,9)(4,6)(5,12)(7,10)(8,13)(11,14)(15,16)(17,18,23,21)(19,24,20,27)(22,29,30,25)(26,32,28,31)(33,34,39,40)(35,37,46,41)(36,47,48,42)(38,53,54,43)(44,55,49,50)(45,56,59,52)(51,57,61,60)(58,63,62,64);(2,4)(3,10)(8,12)(11,16)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32)(34,40)(36,48)(37,41)(43,53)(44,49)(51,61)(52,56)(58,62);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,20,23,19)(18,24,21,27)(22,28,30,26)(25,31,29,32)(33,35)(34,41)(36,49)(37,40)(38,45)(39,46)(42,50)(43,56)(44,48)(47,55)(51,62)(52,53)(54,59)(57,63)(58,61)(60,64);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,21,23,18)(19,24,20,27)(22,29,30,25)(26,31,28,32)(33,36,39,48)(34,42,40,47)(35,44,46,49)(37,50,41,55)(38,51,54,61)(43,57,53,60)(45,58,59,62)(52,63,56,64);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,37,39,41)(34,35,40,46)(36,50,48,55)(38,52,54,56)(42,44,47,49)(43,45,53,59)(51,63,61,64)(57,58,60,62);(33,38,39,54)(34,43,40,53)(35,45,46,59)(36,51,48,61)(37,52,41,56)(42,57,47,60)(44,58,49,62)(50,63,55,64)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022630);