#10142137

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=ghi,e^2=gh,f^2=g,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, 10049333);