#9713391

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

3799 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)(3,13)(4,10)(5,11)(6,16)(7,9)(8,14)(12,15)(17,18)(19,26)(20,23)(21,24)(22,28)(25,30)(27,31)(29,32)(34,36)(35,38)(41,44)(43,45);(1,3,7,14)(2,8,9,13)(4,12,11,16)(5,15,10,6)(19,21)(24,26)(27,29)(31,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,9)(4,11)(8,13)(12,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,11)(2,10,9,5)(3,12,14,16)(6,13,15,8)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,5,7,10)(2,11,9,4)(3,6,14,15)(8,12,13,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6)(2,12)(3,5)(4,13)(7,15)(8,11)(9,16)(10,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=gi,f^2=ghi,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, 9620587);