#10166160

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

3988 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18,22,23)(19,27,28,24)(20,25,29,32)(21,30,31,26)(33,34)(35,45)(36,42)(37,38)(39,43)(40,44)(41,48)(46,47);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(18,23)(20,29)(24,27)(26,30)(33,35,40,48)(34,41,44,45)(36,46,37,39)(38,47,42,43);(2,9)(3,13)(4,15)(11,16)(17,19,22,28)(18,24,23,27)(20,26,29,30)(21,25,31,32)(33,36,40,37)(34,42,44,38)(35,46,48,39)(41,43,45,47);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,19)(18,24)(20,30)(21,25)(22,28)(23,27)(26,29)(31,32)(33,37,40,36)(34,42,44,38)(35,39,48,46)(41,43,45,47);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,20)(18,25)(19,26)(21,27)(22,29)(23,32)(24,31)(28,30)(33,38,40,42)(34,37,44,36)(35,47,48,43)(39,45,46,41);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,21,22,31)(18,26,23,30)(19,25,28,32)(20,24,29,27)(33,39)(34,43)(35,36)(37,48)(38,41)(40,46)(42,45)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,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, 10073356);