#9039342

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
prime radical	#40803	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|b^2=ghi,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,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946538);