#10201901

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	#48271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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:

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

Polycyclic presentation:

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

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, 10109097);