#9481441

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

4069 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(19,25)(23,30)(26,28)(31,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,48)(44,45)(46,47);(5,13)(8,15)(10,16)(12,14)(18,22)(23,30)(24,27)(31,32)(33,35,38,40)(34,36,42,39)(37,46,47,43)(41,48,45,44);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,46)(43,45)(47,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,47)(34,41,42,45)(35,43,40,46)(36,44,39,48)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=g,f^2=h,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, 9388637);