#8649946

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(33,34,39,42)(35,40,45,38)(36,46,37,41)(43,47,44,48);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30)(33,35,39,45)(34,40,42,38)(36,44,37,43)(41,47,46,48);(2,8)(5,10)(9,12)(14,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,32,31)(33,36,39,37)(34,41,42,46)(35,43,45,44)(38,47,40,48);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30)(33,37,39,36)(34,41,42,46)(35,44,45,43)(38,47,40,48);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32)(33,34)(35,38)(36,46)(37,41)(39,42)(40,45)(43,48)(44,47);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,21,22,29)(18,24,25,32)(19,27,20,28)(23,30,26,31)(33,38)(34,35)(36,47)(37,48)(39,40)(41,43)(42,45)(44,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557142);