#8709211

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

4023 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(33,34,39,42)(35,40,45,38)(36,37,48,41)(43,47,46,44);(2,3)(7,9)(8,10)(15,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35,39,45)(34,40,42,38)(36,46,48,43)(37,44,41,47);(2,3)(7,9)(8,10)(15,16)(18,23)(19,25)(24,31)(26,32)(33,36,39,48)(34,41,42,37)(35,43,45,46)(38,47,40,44);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36)(34,37)(35,43)(38,44)(39,48)(40,47)(41,42)(45,46);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37)(34,36)(35,44)(38,43)(39,41)(40,46)(42,48)(45,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,38)(34,35)(36,47)(37,46)(39,40)(41,43)(42,45)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,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, 8616407);