#8633558

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	#46481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,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, 8540754);