#10040743

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,24,20,27)(22,29,30,25)(26,32,28,31)(33,34,38,36)(35,40,43,39)(37,44,45,41)(42,46,48,47);(3,9)(7,13)(10,15)(14,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,34)(35,39)(36,38)(37,41)(40,43)(42,46)(44,45)(47,48);(2,6)(7,13)(8,11)(14,16)(18,21)(19,20)(25,29)(26,28)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,24,21,27)(22,28,30,26)(25,31,29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,24,20,27)(22,25,30,29)(26,32,28,31)(33,36,38,34)(35,39,43,40)(37,41,45,44)(42,47,48,46);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22,23,30)(18,25,21,29)(19,26,20,28)(24,31,27,32)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947939);