#10128126

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	#51166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=ghi,e^2=g,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, 10035322);