#8709192

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

4178 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(34,40)(35,36)(42,47)(43,44);(4,13)(8,12)(11,14)(15,16)(18,23)(19,25)(24,31)(26,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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, 8616388);