#8158935

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,10)(5,14)(11,13)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,42)(35,40,36,39)(37,45,46,41)(43,47,44,48);(1,2,6,7)(3,8,10,15)(4,5,9,14)(11,16,13,12)(18,22)(23,25)(24,29)(30,31)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(3,10)(8,15)(11,13)(12,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,15)(9,11)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,4,6,9)(2,5,7,14)(3,11,10,13)(8,16,15,12)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,5)(2,9)(3,12)(4,7)(6,14)(8,11)(10,16)(13,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,e^2=g,f^2=h,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, 8066131);