#8602456

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

4082 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8509652);