#8343519

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	#37515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,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, 8250715);