#8115446

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	#53134	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,28)(20,21)(22,24)(23,25)(26,27)(29,32)(30,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,6)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,25,28)(20,31,27,29)(21,32,26,30)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(3,10)(7,9)(11,16)(14,15)(18,25)(20,27)(24,28)(29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,32,28,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,31,28,29)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,32,26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022642);