#9685516

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	#54572	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,30)(27,31)(29,32)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(2,7)(4,8)(9,13)(12,14)(17,19,22,21)(18,23,25,26)(20,27,30,29)(24,31,28,32)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(2,7)(3,10)(9,13)(11,16)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21,22,19)(18,23,25,26)(20,29,30,27)(24,31,28,32)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)

Polycyclic presentation:

2^9|a^2=i,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, 9592712);