#8161256

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

3946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,8)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,31,28,25)(37,45)(40,47)(43,46)(44,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19,22,29)(18,23,26,27)(20,28,30,21)(24,31,32,25)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,8)(7,9)(10,13)(15,16)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,35,38,39)(34,36,41,42)(37,43,45,46)(40,44,47,48);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,21)(18,25)(19,20)(22,28)(23,24)(26,31)(27,32)(29,30)(33,36,38,42)(34,39,41,35)(37,44,45,48)(40,46,47,43);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,27)(26,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8068452);