#8898569

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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)(3,10)(4,9)(5,11)(6,16)(7,8)(12,15)(13,14)(17,18)(19,21)(20,27)(22,30)(23,25)(24,31)(26,28)(29,32)(33,34,39,38)(35,41,45,40)(36,46,44,42)(37,43,48,47);(2,8)(5,16)(9,10)(12,14)(17,19,23,27)(18,21,25,20)(22,26,31,32)(24,29,30,28)(33,35,39,45)(34,40,38,41)(36,37,44,48)(42,43,46,47);(1,3,7,4)(2,9,8,10)(5,12,16,14)(6,15,11,13)(17,20,23,21)(18,19,25,27)(22,29,31,28)(24,32,30,26)(33,35,39,45)(34,41,38,40)(36,37,44,48)(42,47,46,43);(1,4,7,3)(2,10,8,9)(5,12,16,14)(6,15,11,13)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,36)(34,42)(35,37)(38,46)(39,44)(40,43)(41,47)(45,48);(1,5)(2,11)(3,12)(4,14)(6,8)(7,16)(9,13)(10,15)(17,19)(18,21)(20,25)(22,26)(23,27)(24,29)(28,30)(31,32)(33,37)(34,43)(35,44)(36,45)(38,47)(39,48)(40,46)(41,42);(1,6,7,11)(2,5,8,16)(3,13,4,15)(9,14,10,12)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,30)(27,32)(33,38,39,34)(35,41,45,40)(36,46,44,42)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805765);