#8821924

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

4068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729120);