#9685539

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	#48111	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9592735);