#8115427

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3656 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)(33,34,38,39)(35,43,46,40)(36,47,48,41)(37,51,52,42)(44,53,49,55)(45,54,59,57)(50,56,61,60)(58,64,62,63);(2,4)(3,10)(8,12)(11,16)(17,19,23,20)(18,24,21,27)(22,26,30,28)(25,31,29,32)(34,39)(36,48)(40,43)(42,51)(44,49)(50,61)(54,57)(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)(34,40)(36,49)(37,45)(38,46)(39,43)(41,55)(42,54)(44,48)(47,53)(50,62)(51,57)(52,59)(56,64)(58,61)(60,63);(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)(33,36,38,48)(34,41,39,47)(35,44,46,49)(37,50,52,61)(40,53,43,55)(42,56,51,60)(45,58,59,62)(54,63,57,64);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,53)(45,54)(47,48)(49,55)(50,56)(51,52)(57,59)(58,63)(60,61)(62,64);(33,37,38,52)(34,42,39,51)(35,45,46,59)(36,50,48,61)(40,54,43,57)(41,56,47,60)(44,58,49,62)(53,63,55,64)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,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, 8022623);