#8926584

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	#58489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3831 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833780);