#8583473

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	#36339	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4152 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)(33,34,40,46)(35,44,38,41)(36,43,37,42)(39,45,48,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32)(33,35)(34,41)(36,45)(37,47)(38,40)(39,43)(42,48)(44,46);(5,14)(8,12)(10,13)(15,16)(18,23)(20,30)(24,32)(26,27)(33,36,40,37)(34,42,46,43)(35,45,38,47)(39,41,48,44);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31)(33,37,40,36)(34,43,46,42)(35,45,38,47)(39,41,48,44);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,38,40,35)(34,44,46,41)(36,45,37,47)(39,42,48,43);(1,3)(2,9)(4,7)(5,10)(6,11)(8,16)(12,15)(13,14)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,39)(34,45)(35,43)(36,41)(37,44)(38,42)(40,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,c^2=h,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, 8490669);