#8963634

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870830);