#8389285

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	#36887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296481);