#8993664

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900860);