#9685527

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 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:

(2,6)(7,13)(8,14)(15,16)(17,18)(19,28)(20,29)(21,27)(22,24)(23,25)(26,30)(31,32)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,29,32,26)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(18,25)(20,30)(24,28)(27,31)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,20)(18,26)(19,27)(21,24)(22,31)(23,30)(25,29)(28,32)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21)(18,27)(19,26)(20,24)(22,29)(23,32)(25,31)(28,30)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,26,32,29)(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=h,c^2=i,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, 9592723);