#8709147

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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, 8616343);