#10106570

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,21)(19,27,29,24)(20,30,28,25)(22,26,32,31)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,41,45,47);(4,11)(8,14)(10,15)(12,16)(17,19,23,29)(18,24,21,27)(20,22,28,32)(25,26,30,31)(35,44)(37,45)(39,43)(41,47);(2,7)(3,9)(8,14)(10,15)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,28)(18,25,21,30)(19,22,29,32)(24,26,27,31)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,47,43,41);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,18)(19,27,29,24)(20,25,28,30)(22,26,32,31)(33,37)(34,41)(35,36)(38,45)(39,48)(40,43)(42,47)(44,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,26)(19,28)(20,29)(21,31)(23,32)(24,30)(25,27)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^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, 10013766);