#8108778

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	#36927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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)(8,13)(10,15)(18,23)(19,21)(20,28)(31,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,48)(44,47)(45,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,18,22,23)(19,24,21,26)(20,29,28,25)(27,32,30,31)(34,39)(40,46)(41,42)(47,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19,22,21)(18,24,23,26)(20,27,28,30)(25,31,29,32)(33,35,38,45)(34,40,39,46)(36,43,37,44)(41,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,36,38,37)(34,41,39,42)(35,43,45,44)(40,47,46,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20,22,28)(18,25,23,29)(19,27,21,30)(24,31,26,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,19)(18,26,23,24)(20,30,28,27)(25,32,29,31)(33,37,38,36)(34,42,39,41)(35,44,45,43)(40,48,46,47)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=h,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, 8015974);