#8158794

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8065990);