#8720428

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(10,14)(11,16)(13,15)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32)(33,34)(35,39)(36,43)(37,42)(38,41)(40,44)(45,47)(46,48);(2,3)(7,9)(8,10)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,40,46)(34,39,44,48)(36,37,38,45)(41,47,43,42);(2,3)(7,9)(8,10)(15,16)(20,29)(23,28)(26,32)(30,31)(33,36,40,38)(34,41,44,43)(35,45,46,37)(39,42,48,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,21,29)(18,23,25,28)(19,26,27,32)(22,30,24,31)(33,37,40,45)(34,42,44,47)(35,36,46,38)(39,43,48,41);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,38,40,36)(34,43,44,41)(35,45,46,37)(39,42,48,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,18)(19,22)(20,28)(21,25)(23,29)(24,27)(26,31)(30,32)(33,39,40,48)(34,35,44,46)(36,42,38,47)(37,41,45,43)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627624);