#10521331

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

4147 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)(3,8)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,27)(26,28)(29,32)(34,40)(35,36)(38,42)(45,47);(2,7)(3,10)(9,15)(11,16)(17,19,23,29)(18,24,27,32)(20,28,21,22)(25,31,30,26)(33,34)(35,44)(36,41)(37,42)(38,43)(39,40)(45,46)(47,48);(17,20,23,21)(18,25,27,30)(19,28,29,22)(24,31,32,26)(33,35,39,36)(34,41,40,44)(37,45,43,47)(38,46,42,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,21,23,20)(18,25,27,30)(19,22,29,28)(24,31,32,26)(33,36,39,35)(34,41,40,44)(37,45,43,47)(38,48,42,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32)(33,37,39,43)(34,42,40,38)(35,45,36,47)(41,48,44,46);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,22)(18,26)(19,20)(21,29)(23,28)(24,25)(27,31)(30,32)(33,38,39,42)(34,43,40,37)(35,46,36,48)(41,47,44,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428527);