#10143669

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(19,24)(20,26)(21,27)(22,30)(23,25)(28,31)(29,32)(34,39)(35,41)(37,45)(44,46);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,28)(18,24,25,31)(20,29,30,21)(22,27,26,32)(33,34,38,39)(35,40,41,36)(37,42,45,47)(43,46,48,44);(18,25)(19,28)(20,30)(27,32)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,28)(22,25)(23,30)(24,32)(27,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,46)(44,47)(45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,23,29)(18,27,25,32)(19,30,28,20)(22,31,26,24)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22,23,26)(18,20,25,30)(19,27,28,32)(21,24,29,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,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, 10050865);