#8720717

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

4199 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=gi,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, 8627913);