#8159199

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

4078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=gh,f^2=gh,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, 8066395);