#8602809

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8510005);