#9713387

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

3805 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,28)(20,26)(21,27)(22,24)(23,25)(29,32)(30,31)(35,38)(36,40)(43,47)(44,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,22)(18,24,25,28)(20,29,30,27)(21,31,32,26)(33,34)(35,36)(37,45)(38,40)(39,42)(41,46)(43,48)(44,47);(2,7)(3,9)(8,14)(10,15)(18,25)(20,30)(24,28)(27,29)(33,35,39,38)(34,40,42,36)(37,43,46,47)(41,48,45,44);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,30)(18,26,25,31)(19,29,22,27)(21,28,32,24)(33,36,39,40)(34,35,42,38)(37,44,46,48)(41,43,45,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21)(18,27)(19,26)(20,28)(22,31)(23,32)(24,30)(25,29)(33,37)(34,41)(35,43)(36,44)(38,47)(39,46)(40,48)(42,45);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,19)(18,28,25,24)(20,27,30,29)(21,26,32,31)(33,38,39,35)(34,40,42,36)(37,43,46,47)(41,44,45,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=ghi,f^2=gi,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, 9620583);