#8550899

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

4085 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8458095);