#8739186

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18)(19,26)(20,22)(21,25)(23,24)(27,28)(29,32)(30,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,6,12)(2,7,10,11)(4,9,13,14)(5,8,15,16)(18,24)(20,29)(25,26)(27,30)(34,35)(39,41)(40,42)(46,47);(1,4,6,13)(2,8,10,16)(3,9,12,14)(5,7,15,11)(17,19,23,21)(18,25,24,26)(20,30,29,27)(22,28,32,31)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,8)(3,9)(5,11)(6,13)(7,15)(10,16)(12,14)(17,20,23,29)(18,22,24,32)(19,27,21,30)(25,31,26,28)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,21,23,19)(18,26,24,25)(20,30,29,27)(22,28,32,31);(1,5,6,15)(2,9,10,14)(3,8,12,16)(4,7,13,11)(17,22)(18,20)(19,28)(21,31)(23,32)(24,29)(25,30)(26,27)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=h,f^2=gh,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, 8646382);