#10423005

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

4280 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10330201);