#8533305

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	#52316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4232 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)(3,8)(4,10)(5,6)(7,9)(11,13)(12,14)(15,16)(17,18)(19,27)(20,29)(21,23)(22,26)(24,28)(25,31)(30,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(1,3,7,12)(2,8,9,14)(4,13,15,5)(6,10,11,16)(17,19,23,28)(18,24,21,27)(20,30,31,22)(25,32,29,26)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,9)(3,12)(5,13)(10,16)(18,21)(24,27)(25,29)(26,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,4)(2,10)(3,5)(6,8)(7,15)(9,16)(11,14)(12,13)(17,20,23,31)(18,25,21,29)(19,22,28,30)(24,26,27,32)(33,36,39,35)(34,41,37,40)(38,43,46,44)(42,47,45,48);(1,5)(2,11)(3,4)(6,9)(7,13)(8,16)(10,14)(12,15)(17,21,23,18)(19,27,28,24)(20,29,31,25)(22,32,30,26)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,6,7,11)(2,5,9,13)(3,10,12,16)(4,14,15,8)(17,22)(18,26)(19,20)(21,32)(23,30)(24,25)(27,29)(28,31)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 8440501);