#9007354

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	#42409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8914550);