#8630812

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,42)(36,44)(37,39)(38,40)(41,45)(43,47)(46,48);(1,3,7,11)(2,8,5,15)(4,12,14,10)(6,9,16,13)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(34,37)(36,45)(40,42)(43,46);(2,5)(3,11)(9,13)(10,12)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,35,39,38)(34,40,37,42)(36,43,45,46)(41,47,44,48);(1,4,7,14)(2,9,5,13)(3,10,11,12)(6,15,16,8)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,5,7,2)(3,8,11,15)(4,13,14,9)(6,12,16,10)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,39,34)(35,40,38,42)(36,44,45,41)(43,47,46,48);(1,6)(2,10)(3,9)(4,8)(5,12)(7,16)(11,13)(14,15)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,39,35)(34,40,37,42)(36,43,45,46)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538008);