#8934833

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	#50285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,8)(4,10)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,39,42)(35,40,43,48)(36,46,45,38)(37,47,44,41);(2,7)(3,10)(9,13)(11,16)(17,18,21,22)(19,23,27,25)(20,24,29,28)(26,30,32,31)(33,35)(34,40)(36,37)(38,41)(39,43)(42,48)(44,45)(46,47);(2,7)(3,10)(5,14)(12,15)(18,22)(20,29)(23,25)(26,32)(35,43)(36,45)(38,46)(40,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36)(34,38)(35,44)(37,43)(39,45)(40,47)(41,48)(42,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37,39,44)(34,41,42,47)(35,45,43,36)(38,40,46,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38)(34,36)(35,41)(37,40)(39,46)(42,45)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 8842029);