#8159541

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

4136 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)(3,5)(4,9)(6,14)(7,8)(10,15)(11,16)(12,13)(21,29)(24,30)(26,31)(27,32)(34,39)(35,43)(37,41)(44,46);(1,3)(2,5)(4,7)(6,11)(8,9)(10,13)(12,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(2,8)(3,4)(10,14)(11,12)(18,23)(19,25)(24,30)(26,31)(35,43)(40,47)(44,46)(45,48);(1,4,7,3)(2,9,8,5)(6,12,15,11)(10,16,14,13)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,35)(34,40)(36,46)(37,45)(38,43)(39,47)(41,48)(42,44);(1,5,7,9)(2,4,8,3)(6,13,15,16)(10,12,14,11)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,36,38,42)(34,41,39,37)(35,44,43,46)(40,45,47,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,41)(34,42,39,36)(35,45,43,48)(40,44,47,46)

Polycyclic presentation:

2^9|d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066737);