#8696787

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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)(3,7,10,14)(4,8,13,11)(9,15,12,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,6)(7,14)(8,11)(15,16)(18,24)(19,20)(26,29)(27,28)(35,41)(38,44)(42,47)(45,48);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603983);