#9685565

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

3987 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,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, 9592761);