#8183770

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	#59228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(18,21)(19,20)(26,29)(27,28)(33,34,40,39)(35,45,47,41)(36,48,46,42)(37,44,38,43);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35)(34,41)(36,37)(38,46)(39,45)(40,47)(42,44)(43,48);(2,6)(7,13)(8,11)(14,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36,40,46)(34,42,39,48)(35,37,47,38)(41,44,45,43);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,37,40,38)(34,43,39,44)(35,36,47,46)(41,48,45,42);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,38,40,37)(34,44,39,43)(35,46,47,36)(41,42,45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,39,40,34)(35,45,47,41)(36,42,46,48)(37,44,38,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8090966);