#8118863

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	#47172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(21,29)(24,31)(27,28)(30,32)(33,34)(35,41)(36,37)(38,40)(39,43)(42,47)(44,48)(45,46);(2,3)(7,9)(8,10)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35,39,38)(34,40,43,41)(36,44,47,46)(37,45,42,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32)(33,35,39,38)(34,41,43,40)(36,44,47,46)(37,48,42,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32)(33,36)(34,42)(35,44)(37,43)(38,46)(39,47)(40,48)(41,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,37,39,42)(34,36,43,47)(35,45,38,48)(40,44,41,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,35)(34,41,43,40)(36,46,47,44)(37,48,42,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=i,e^2=g,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, 8026059);