#9481468

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

3905 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,24)(19,25,20,26)(21,27,32,22)(28,29,30,31)(34,38)(36,44)(39,41)(42,48);(3,9)(7,13)(10,15)(14,16)(18,24)(21,32)(25,26)(28,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,6)(7,13)(8,11)(14,16)(17,19,23,20)(18,25,24,26)(21,28,32,30)(22,31,27,29)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,26,24,25)(21,28,32,30)(22,31,27,29)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21)(18,27)(19,28)(20,30)(22,24)(23,32)(25,29)(26,31)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,27)(18,21,24,32)(19,29,20,31)(25,30,26,28)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gi,e^2=h,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, 9388664);