#8322282

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

4076 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,18,22,21)(19,25,27,23)(20,24,30,28)(26,32,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(2,8)(5,7)(9,12)(14,16)(18,21)(23,25)(24,28)(31,32)(34,39)(37,47)(40,45)(41,48);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,35)(34,40)(36,42)(37,41)(38,44)(39,45)(43,46)(47,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,48)(40,47)(42,44);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(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=gh,d^2=g,e^2=gh,f^2=gi,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, 8229478);