#9007824

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8915020);