#8720498

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,20)(19,24,25,21)(22,26,31,28)(27,30,32,29)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19,23,25)(18,24,20,21)(22,27,31,32)(26,30,28,29)(36,44)(41,46)(43,45)(47,48);(4,13)(7,12)(10,16)(14,15)(18,20)(19,25)(26,28)(27,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,18)(19,24,25,21)(22,26,31,28)(27,29,32,30)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21,23,24)(18,25,20,19)(22,30,31,29)(26,27,28,32)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,2)(3,6)(4,12)(5,9)(7,13)(8,11)(10,15)(14,16)(17,22,23,31)(18,26,20,28)(19,27,25,32)(21,29,24,30)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,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, 8627694);