#8946220

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

4132 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8853416);