#8159281

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	#41466	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=h,f^2=gh,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, 8066477);