#10247643

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

3705 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:

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

Polycyclic presentation:

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

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, 10154839);