#10195716

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=hi,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, 10102912);