#9712861

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=h,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, 9620057);