#8622273

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	#41120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529469);