#8739192

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,18)(19,25)(20,31)(21,27)(22,32)(23,28)(24,29)(26,30)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19,24,22)(18,25,29,32)(20,23,30,27)(21,31,28,26)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,20)(18,26)(19,23)(21,25)(22,27)(24,30)(28,32)(29,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,2)(3,6)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,21,24,28)(18,27,29,23)(19,26,22,31)(20,25,30,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,22,24,19)(18,25,29,32)(20,27,30,23)(21,31,28,26);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,23,24,27)(18,28,29,21)(19,30,22,20)(25,26,32,31)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,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, 8646388);