#8317527

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

4055 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8224723);