#10247638

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

3711 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,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, 10154834);