#8718326

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=h,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, 8625522);