#8425486

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3949 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,24,23)(19,28,27,22)(20,30,32,25)(21,29,31,26)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(2,3)(7,9)(8,10)(14,15)(17,19)(18,22)(20,29)(21,30)(23,28)(24,27)(25,31)(26,32)(34,39)(35,42)(41,44)(43,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20)(18,25)(19,29)(21,28)(22,31)(23,30)(24,32)(26,27)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,21,24,31)(18,26,23,29)(19,25,27,30)(20,28,32,22)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,22,24,28)(18,27,23,19)(20,31,32,21)(25,26,30,29);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,23,24,18)(19,28,27,22)(20,30,32,25)(21,26,31,29)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332682);