#9724836

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	#50843	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(6,10)(7,8)(9,15)(11,16)(12,14)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34)(35,41)(36,47)(37,40)(38,42)(39,45)(43,48)(44,46);(1,3,6,8)(2,7,10,4)(5,11,15,14)(9,16,13,12)(17,19)(18,24)(20,22)(21,30)(23,31)(25,32)(26,27)(28,29)(33,35,40,46)(34,41,37,44)(36,45,38,43)(39,42,48,47);(1,4,6,7)(2,8,10,3)(5,12,15,16)(9,14,13,11)(17,20,23,21)(18,25,28,26)(19,22,31,30)(24,32,29,27)(33,36,40,38)(34,42,37,47)(35,43,46,45)(39,44,48,41);(1,4)(2,3)(5,12)(6,7)(8,10)(9,11)(13,14)(15,16)(17,21,23,20)(18,26,28,25)(19,30,31,22)(24,27,29,32)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,45,48,43);(1,3)(2,4)(5,14)(6,8)(7,10)(9,16)(11,15)(12,13)(17,22,23,30)(18,27,28,32)(19,20,31,21)(24,26,29,25)(33,38,40,36)(34,42,37,47)(35,45,46,43)(39,44,48,41);(1,5)(2,9)(3,11)(4,12)(6,15)(7,16)(8,14)(10,13)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,39,40,48)(34,43,37,45)(35,42,46,47)(36,44,38,41)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9632032);