#8720684

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

4226 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627880);