#8121434

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	#40272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8028630);