#8709209

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=g,e^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, 8616405);