#8406460

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	#2815	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(34,40)(35,44)(36,48)(42,46);(4,12)(7,14)(10,16)(13,15)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=gh,f^2=h,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, 8313656);