#8534539

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	#54839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=g,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, 8441735);