#8505360

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,23,20)(19,24,29,27)(21,30,32,25)(22,26,28,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(2,4)(7,9)(8,12)(15,16)(17,19)(18,24)(20,27)(21,22)(23,29)(25,31)(26,30)(28,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(17,18,23,20)(19,27,29,24)(21,25,32,30)(22,26,28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,20,23,18)(19,27,29,24)(21,30,32,25)(22,26,28,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,21)(18,25)(19,28)(20,30)(22,29)(23,32)(24,26)(27,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,22,23,28)(18,26,20,31)(19,21,29,32)(24,30,27,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412556);