#8755187

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=ghi,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, 8662383);