#8108781

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	#40007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8015977);