#8934889

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	#47812	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842085);