#8734604

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

3954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641800);