#10181374

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

3833 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=i,e^2=ghi,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, 10088570);