#8602464

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	#52375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34,40,45)(35,38,46,41)(36,43,37,42)(39,44,47,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28)(33,35,40,46)(34,41,45,38)(36,39,37,47)(42,44,43,48);(5,14)(8,12)(10,13)(15,16)(18,24)(19,27)(21,32)(26,28)(33,36,40,37)(34,42,45,43)(35,39,46,47)(38,48,41,44);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31)(33,37,40,36)(34,43,45,42)(35,39,46,47)(38,48,41,44);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31)(33,38,40,41)(34,35,45,46)(36,44,37,48)(39,42,47,43);(1,3)(2,9)(4,7)(5,10)(6,11)(8,16)(12,15)(13,14)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)(33,39)(34,44)(35,37)(36,46)(38,42)(40,47)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^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, 8509660);