#10171819

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	#46393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=gi,f^2=g,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, 10079015);