#8343521

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

3985 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,18,22,21)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,35)(34,41)(36,46)(37,43)(38,47)(39,42)(40,48)(44,45);(2,9)(6,16)(8,11)(10,14)(18,21)(23,25)(24,28)(31,32)(33,36,40,37)(34,42,44,38)(35,46,48,43)(39,45,47,41);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,37,40,36)(34,42,44,38)(35,43,48,46)(39,45,47,41);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,38,40,42)(34,37,44,36)(35,47,48,39)(41,43,45,46);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,39,40,47)(34,43,44,46)(35,42,48,38)(36,41,37,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8250717);