#8517553

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(36,44)(39,46)(42,48)(45,47);(5,13)(8,12)(10,16)(14,15)(18,23)(20,30)(24,32)(26,27)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424749);