#8581815

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3820 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=i,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8489011);