#8720407

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627603);