#9685569

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(2,7)(4,14)(8,11)(10,13)(17,19,23,20)(18,24,22,25)(21,27,29,28)(26,31,30,32)(34,40)(35,37)(42,46)(43,45);(2,7)(3,12)(9,16)(10,13)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,12)(2,8,7,11)(4,13,14,10)(5,9,15,16)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=g,e^2=h,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, 9592765);