#8284484

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191680);