#9713354

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3759 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)(3,6)(4,5)(7,15)(8,10)(9,16)(11,12)(13,14)(19,22)(20,24)(27,31)(28,32)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(1,3,8,5)(2,4,10,6)(7,11,16,13)(9,12,15,14)(17,18)(19,20)(21,29)(22,24)(23,26)(25,30)(27,32)(28,31)(33,35,38,45)(34,39,41,42)(36,43,40,47)(37,44,46,48);(1,4,8,6)(2,3,10,5)(7,12,16,14)(9,11,15,13)(17,19,23,22)(18,24,26,20)(21,27,30,31)(25,32,29,28);(1,5,8,3)(2,4,10,6)(7,13,16,11)(9,12,15,14)(17,20,23,24)(18,19,26,22)(21,28,30,32)(25,27,29,31);(1,6,8,4)(2,3,10,5)(7,12,16,14)(9,13,15,11)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,32)(26,29)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,7)(2,9)(3,11)(4,12)(5,13)(6,14)(8,16)(10,15)(17,22,23,19)(18,24,26,20)(21,27,30,31)(25,28,29,32)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9620550);