#8963682

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870878);