#8323156

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230352);