#8934862

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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)(3,7)(4,5)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,29)(21,26)(22,27)(24,28)(25,31)(30,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,3,6,12)(2,7,8,9)(4,13,11,15)(5,14,10,16)(17,19,22,28)(18,23,27,24)(20,30,31,26)(21,29,32,25)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(2,8)(3,12)(5,10)(13,15)(19,28)(21,32)(23,24)(26,30)(34,36)(38,46)(40,41)(43,45);(1,3)(2,9)(4,15)(5,14)(6,12)(7,8)(10,16)(11,13)(17,19)(18,24)(20,30)(21,29)(22,28)(23,27)(25,32)(26,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,11)(2,10,8,5)(3,13,12,15)(7,16,9,14)(17,20,22,31)(18,25,27,29)(19,26,28,30)(21,23,32,24)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,10)(2,11,8,4)(3,14,12,16)(7,15,9,13)(17,21)(18,26)(19,29)(20,23)(22,32)(24,31)(25,28)(27,30)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842058);