#10143430

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	#58553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

(2,7)(3,9)(5,13)(12,14)(17,18)(19,27)(20,25)(21,30)(22,24)(23,26)(28,32)(29,31)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(1,2,6,7)(3,8,9,4)(5,10,13,15)(11,14,16,12)(17,19,23,22)(18,24,26,27)(20,28,21,29)(25,31,30,32)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,14,15,12)(17,20,23,21)(18,25,26,30)(19,28,22,29)(24,31,27,32)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,15)(13,16)(17,21,23,20)(18,25,26,30)(19,29,22,28)(24,31,27,32)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,4,6,8)(2,9,7,3)(5,14,13,12)(10,11,15,16)(17,18)(19,24)(20,25)(21,30)(22,27)(23,26)(28,31)(29,32)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,15)(8,14)(9,16)(17,22,23,19)(18,24,26,27)(20,29,21,28)(25,31,30,32)(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=gi,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, 10050626);