#8755109

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	#55349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662305);