#8741950

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(34,36)(39,41)(40,44)(46,47);(2,7)(3,11)(9,12)(10,16)(18,24)(19,20)(26,29)(27,28)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(3,11)(9,12)(10,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8649146);