#8551224

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	#51687	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=g,e^2=g,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, 8458420);