#10166037

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073233);