#8750751

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=gi,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, 8657947);