#8750491

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4013 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)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18,23,24)(19,27,21,25)(20,26,30,22)(28,32,31,29)(33,34,39,38)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(18,24)(20,30)(25,27)(28,31)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(3,11)(4,13)(7,14)(8,12)(17,19,23,21)(18,25,24,27)(20,31,30,28)(22,29,26,32)(33,34,39,38)(35,40,36,42)(37,41,45,46)(43,47,44,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,20,23,30)(18,26,24,22)(19,28,21,31)(25,29,27,32)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,21,23,19)(18,27,24,25)(20,31,30,28)(22,29,26,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,31,27,28)(33,38,39,34)(35,42,36,40)(37,41,45,46)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=gi,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, 8657687);