#10166317

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3924 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=h,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, 10073513);