#10040588

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	#53749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9947784);