#8726461

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	#59952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=gi,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, 8633657);