#8639924

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,22,23)(19,21,25,27)(20,24,30,32)(26,31,28,29)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(2,4)(3,10)(8,12)(11,13)(17,19)(18,21)(20,28)(22,25)(23,27)(24,29)(26,30)(31,32)(35,41)(39,43)(42,47)(46,48);(2,4)(7,9)(8,12)(15,16)(18,23)(19,25)(24,32)(26,28)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,20,22,30)(18,24,23,32)(19,26,25,28)(21,31,27,29)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,18)(19,27)(20,24)(21,25)(22,23)(26,29)(28,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,27)(18,25,23,19)(20,29,30,31)(24,26,32,28)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8547120);