#8755403

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=ghi,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, 8662599);