#8389277

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

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^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, 8296473);