#9478215

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

4244 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,25)(21,29)(22,24)(26,30)(27,28)(31,32)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,35,39,46)(34,40,36,43)(37,38,45,44)(41,48,47,42);(2,6)(7,13)(8,11)(14,16)(18,24)(19,25)(21,30)(28,31)(33,34,39,36)(35,40,46,43)(37,41,45,47)(38,48,44,42);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,31)(30,32)(33,36,39,34)(35,43,46,40)(37,41,45,47)(38,48,44,42);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20)(18,25)(19,24)(21,28)(22,23)(26,32)(27,29)(30,31)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9385411);