#8183787

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	#57556	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)(17,18)(19,29)(20,26)(21,31)(22,27)(23,25)(24,28)(30,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19,24,23)(18,25,28,29)(20,30,21,27)(22,31,32,26)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(2,6)(7,13)(8,11)(14,15)(17,20,24,21)(18,26,28,31)(19,30,23,27)(22,29,32,25)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,21,24,20)(18,26,28,31)(19,30,23,27)(22,25,32,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,22,24,32)(18,27,28,30)(19,31,23,26)(20,29,21,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,23,24,19)(18,25,28,29)(20,30,21,27)(22,26,32,31)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=gh,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, 8090983);