#8720312

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 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)(3,11,12,8)(4,13,14,9)(5,10,15,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(35,41)(39,43)(42,47)(46,48);(2,7)(3,12)(9,13)(10,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,34,38,37)(35,39,41,43)(36,44,45,40)(42,48,47,46);(2,7)(3,12)(9,13)(10,16)(17,18,21,25)(19,23,27,22)(20,24,30,28)(26,32,29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,9,15,13)(17,18)(19,23)(20,28)(21,25)(22,27)(24,30)(26,31)(29,32)(33,36,38,45)(34,40,37,44)(35,42,41,47)(39,46,43,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,8,15,11)(17,19)(18,23)(20,29)(21,27)(22,25)(24,31)(26,30)(28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,20,21,30)(18,24,25,28)(19,26,27,29)(22,31,23,32)(33,37,38,34)(35,43,41,39)(36,44,45,40)(42,48,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627508);