#8534572

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,8)(4,10)(5,14)(6,11)(7,9)(12,16)(13,15)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3,7,5)(2,8,9,14)(4,12,13,11)(6,10,16,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(34,38)(35,42)(40,44)(43,48);(2,9)(3,5)(10,15)(11,12)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,4,7,13)(2,10,9,15)(3,11,5,12)(6,14,16,8)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,5,7,3)(2,8,9,14)(4,12,13,11)(6,15,16,10)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,6)(2,11)(3,10)(4,8)(5,15)(7,16)(9,12)(13,14)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8441768);