#9706333

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9613529);