#9007279

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

3855 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,e^2=i,f^2=h,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, 8914475);