#9039348

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	#40953	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(27,28)(29,31)(30,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,22,26)(19,23,29,31)(20,21,25,24)(27,32,30,28)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,6)(4,12)(7,10)(11,15)(20,25)(21,24)(27,30)(28,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,19,22,29)(18,23,26,31)(20,30,25,27)(21,28,24,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,25)(18,24,26,21)(19,27,29,30)(23,28,31,32)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,21,22,24)(18,25,26,20)(19,28,29,32)(23,27,31,30)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)

Polycyclic presentation:

2^9|b^2=ghi,d^2=gi,e^2=gh,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, 8946544);