#8901634

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	#52042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808830);