#8126137

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8033333);