#8159127

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066323);