#9015388

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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)(3,4,11,8)(6,13,14,9)(10,15,16,12)(18,23)(19,26)(25,31)(27,29)(33,34)(35,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(33,35,39,47)(34,40,38,46)(36,44,45,37)(41,43,42,48);(2,5)(4,8)(6,14)(10,16)(19,26)(20,28)(21,24)(25,31)(33,36,39,45)(34,41,38,42)(35,37,47,44)(40,48,46,43);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,36)(34,42)(35,44)(37,47)(38,41)(39,45)(40,48)(43,46);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,37,39,44)(34,43,38,48)(35,45,47,36)(40,41,46,42);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(33,38,39,34)(35,46,47,40)(36,41,45,42)(37,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=ghi,e^2=gi,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, 8922584);