#10040671

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,9,10,4)(5,13,15,7)(11,12,16,14)(18,21)(23,25)(24,29)(30,31)(33,34,38,41)(35,40,43,37)(36,39,45,48)(42,44,47,46);(3,10)(4,9)(11,16)(12,14)(20,28)(24,29)(26,32)(30,31)(33,35,38,43)(34,37,41,40)(36,42,45,47)(39,46,48,44);(1,3,6,10)(2,4,8,9)(5,11,15,16)(7,12,13,14)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,36)(34,39)(35,42)(37,46)(38,45)(40,44)(41,48)(43,47);(1,3)(2,4)(5,11)(6,10)(7,12)(8,9)(13,14)(15,16)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31)(33,35)(34,40)(36,42)(37,41)(38,43)(39,44)(45,47)(46,48);(1,4)(2,3)(5,14)(6,9)(7,16)(8,10)(11,13)(12,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,38,40)(34,35,41,43)(36,44,45,46)(39,47,48,42);(1,5,6,15)(2,7,8,13)(3,11,10,16)(4,12,9,14)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)(33,36)(34,39)(35,42)(37,46)(38,45)(40,44)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947867);