#8324360

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	#40036	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,11,12,8)(4,13,15,9)(5,10,16,14)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,39,40)(35,36,41,44)(37,45,47,38)(42,46,48,43);(2,7)(3,12)(9,13)(10,14)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35,39,41)(34,36,40,44)(37,42,47,48)(38,43,45,46);(2,7)(5,16)(8,11)(9,13)(19,25)(22,31)(27,29)(28,32)(34,40)(35,41)(38,45)(42,48);(1,3)(2,8)(4,14)(5,9)(6,12)(7,11)(10,15)(13,16)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36,39,44)(34,41,40,35)(37,46,47,43)(38,42,45,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,8,16,11)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37)(34,38)(35,42)(36,43)(39,47)(40,45)(41,48)(44,46);(1,5,6,16)(2,10,7,14)(3,9,12,13)(4,8,15,11)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,38)(34,37)(35,43)(36,42)(39,45)(40,47)(41,46)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231556);