#8754996

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	#46571	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662192);