#8726509

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	#2537	C₂²
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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633705);