#8158973

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	#38824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,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, 8066169);