#8418178

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	#45160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325374);