#9712193

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,22,24)(19,20,27,23)(21,25,30,32)(26,28,29,31)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,6)(7,9)(8,12)(15,16)(18,24)(20,23)(25,32)(28,31)(33,34,39,37)(35,40,36,41)(38,42,46,45)(43,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9619389);