#9720170

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	#50066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,23)(20,25)(21,31)(22,26)(24,29)(27,28)(30,32)(33,34,40,44)(35,45,48,41)(36,42,37,39)(38,43,46,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,32)(27,31)(33,35,40,48)(34,41,44,45)(36,46,37,38)(39,47,42,43);(2,6)(7,13)(8,11)(14,16)(17,19,22,29)(18,24,26,23)(20,27,31,30)(21,28,25,32)(33,36,40,37)(34,42,44,39)(35,46,48,38)(41,43,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31)(33,37,40,36)(34,39,44,42)(35,38,48,46)(41,47,45,43);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,31)(18,25,26,21)(19,27,29,30)(23,28,24,32)(33,38)(34,43)(35,36)(37,48)(39,41)(40,46)(42,45)(44,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,25)(18,20,26,31)(19,28,29,32)(23,30,24,27)(33,39)(34,36)(35,47)(37,44)(38,45)(40,42)(41,46)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=h,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, 9627366);