#8295658

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	#46107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gi,e^2=h,f^2=g,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, 8202854);