#8317492

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

4240 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=i,f^2=g,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, 8224688);