#8633915

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

4005 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:

(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(34,38)(35,41)(40,46)(43,44);(1,2)(3,8)(4,5)(6,7)(9,13)(10,14)(11,16)(12,15)(20,28)(25,30)(27,29)(31,32)(33,34,37,38)(35,42,41,39)(36,40,45,46)(43,48,44,47);(2,7)(4,13)(8,15)(10,16)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,34,37,38)(35,39,41,42)(36,40,45,46)(43,47,44,48);(1,3)(2,8)(4,10)(5,14)(6,12)(7,15)(9,11)(13,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,35)(34,39)(36,44)(37,41)(38,42)(40,48)(43,45)(46,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,16)(14,15)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,48)(45,46);(1,5,6,9)(2,4,7,13)(3,11,12,14)(8,16,15,10)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(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=g,c^2=h,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, 8541111);