#8633853

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=gh,f^2=hi,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, 8541049);