#10040852

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(19,22)(20,28)(21,26)(23,24)(25,30)(27,32)(29,31)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(4,11)(7,9)(12,16)(14,15)(17,19)(18,22)(20,29)(21,23)(24,26)(25,32)(27,30)(28,31)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(2,3)(7,9)(8,10)(14,15)(18,24)(19,21)(20,30)(31,32)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20)(18,25)(19,27)(21,29)(22,31)(23,30)(24,28)(26,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,19)(18,26,24,22)(20,29,30,27)(25,32,28,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,26)(18,21,24,19)(20,31,30,32)(25,29,28,27)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=ghi,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, 9948048);