#8818152

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

4159 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725348);