#9685562

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	#55457	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,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, 9592758);