#10143426

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	#48018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,e^2=h,f^2=gh,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, 10050622);