#8943956

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
prime radical	#49024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 8851152);