#8709190

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

4188 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)(3,8)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(33,34,40,43)(35,44,39,41)(36,38,37,42)(45,47,46,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,40,39)(34,41,43,44)(36,46,37,45)(38,47,42,48);(4,13)(7,12)(11,16)(14,15)(18,23)(19,25)(24,31)(26,32)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,47,44,48);(1,4,6,13)(2,7,10,12)(3,11,9,16)(5,15,8,14)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,37,40,36)(34,38,43,42)(35,46,39,45)(41,48,44,47);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38)(34,36)(35,47)(37,43)(39,48)(40,42)(41,46)(44,45);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,48,42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,f^2=g,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, 8616386);