#8445935

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353131);