#8746558

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,32)(30,31)(33,34,39,38)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,23,21)(18,24,25,27)(20,28,30,29)(22,26,32,31)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(3,12)(7,8)(13,16)(14,15)(18,25)(20,30)(24,27)(28,29)(33,34,39,38)(35,40,36,42)(37,41,45,46)(43,47,44,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20,23,30)(18,26,25,31)(19,29,21,28)(22,27,32,24)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,19)(18,27,25,24)(20,28,30,29)(22,26,32,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,32)(18,28,25,29)(19,26,21,31)(20,24,30,27)(33,38,39,34)(35,42,36,40)(37,41,45,46)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,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, 8653754);