#10195461

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,39,43)(35,44,42,38)(36,47,46,40)(37,41,45,48);(1,3,7,4)(2,8,6,9)(5,11,13,12)(10,15,14,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35,39,42)(34,38,43,44)(36,45,46,37)(40,48,47,41);(18,25)(19,29)(20,31)(27,32)(33,36,39,46)(34,40,43,47)(35,45,42,37)(38,48,44,41);(1,4,7,3)(2,9,6,8)(5,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,36)(34,40)(35,45)(37,42)(38,48)(39,46)(41,44)(43,47);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,37,39,45)(34,41,43,48)(35,46,42,36)(38,47,44,40);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,38)(34,42)(35,43)(36,48)(37,40)(39,44)(41,46)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=gh,e^2=g,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, 10102657);