#8115375

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

3715 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)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34,38,39)(35,43,46,40)(36,47,48,41)(37,50,51,42)(44,52,57,55)(45,53,59,56)(49,54,61,60)(58,63,64,62);(2,3)(4,11)(8,10)(12,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(34,39)(36,48)(40,43)(42,50)(44,57)(49,61)(53,56)(58,64);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35,38,46)(34,40,39,43)(36,44,48,57)(37,45,51,59)(41,52,47,55)(42,53,50,56)(49,58,61,64)(54,62,60,63);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,55)(45,56)(47,48)(49,54)(50,51)(52,57)(53,59)(58,63)(60,61)(62,64);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,36,38,48)(34,41,39,47)(35,44,46,57)(37,49,51,61)(40,52,43,55)(42,54,50,60)(45,58,59,64)(53,62,56,63);(33,37,38,51)(34,42,39,50)(35,45,46,59)(36,49,48,61)(40,53,43,56)(41,54,47,60)(44,58,57,64)(52,62,55,63)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^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, 8022571);