#8688360

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

4099 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8595556);