#8639868

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,30,29)(20,31,21,25)(22,26,32,28)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(2,4)(3,10)(8,12)(11,13)(17,19,23,30)(18,24,27,29)(20,28,21,26)(22,25,32,31)(35,41)(39,43)(42,47)(46,48);(2,4)(7,9)(8,12)(15,16)(17,20,23,21)(18,25,27,31)(19,26,30,28)(22,29,32,24)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,21,23,20)(18,25,27,31)(19,28,30,26)(22,29,32,24)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,18)(19,29)(20,31)(21,25)(22,26)(23,27)(24,30)(28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22,23,32)(18,26,27,28)(19,25,30,31)(20,29,21,24)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=gi,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, 8547064);