#8720411

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(10,14)(11,16)(13,15)(17,18,23,24)(19,22,25,32)(20,21,29,30)(26,31,28,27)(34,39)(35,42)(36,44)(45,47);(2,3)(7,9)(8,10)(15,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(2,3)(7,9)(8,10)(15,16)(18,24)(19,25)(20,29)(27,31)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,21)(19,26)(22,31)(23,29)(24,30)(25,28)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,31)(26,32)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,32)(18,25,24,19)(20,27,29,31)(21,26,30,28)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627607);