#8114399

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

3707 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)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(37,49)(42,51)(45,59)(48,60)(53,56)(54,61)(58,64)(62,63);(2,7)(3,10)(9,12)(11,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(34,39)(36,47)(40,43)(42,51)(44,57)(48,60)(53,56)(58,64);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,55)(45,56)(47,50)(48,54)(49,51)(52,57)(53,59)(58,63)(60,61)(62,64);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,35,38,46)(34,40,39,43)(36,44,47,57)(37,45,49,59)(41,52,50,55)(42,53,51,56)(48,58,60,64)(54,62,61,63);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(33,36,38,47)(34,41,39,50)(35,44,46,57)(37,48,49,60)(40,52,43,55)(42,54,51,61)(45,58,59,64)(53,62,56,63);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,42)(35,45)(36,48)(38,49)(39,51)(40,53)(41,54)(43,56)(44,58)(46,59)(47,60)(50,61)(52,62)(55,63)(57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gi,e^2=gi,f^2=g,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, 8021595);