#10171707

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=gi,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, 10078903);