#8496490

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403686);