#9206585

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113781);