#8418543

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

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

Polycyclic presentation:

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

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, 8325739);