#10042600

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

4099 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,39,36)(35,43,44,40)(37,41,47,45)(38,42,48,46);(2,7)(3,10)(4,13)(12,15)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35)(34,40)(36,43)(37,38)(39,44)(41,46)(42,45)(47,48);(2,7)(3,10)(4,13)(12,15)(18,25)(19,29)(20,31)(27,32)(33,34,39,36)(35,40,44,43)(37,45,47,41)(38,42,48,46);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,36,39,34)(35,40,44,43)(37,45,47,41)(38,46,48,42);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,37,39,47)(34,41,36,45)(35,38,44,48)(40,46,43,42);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,38)(34,42)(35,37)(36,46)(39,48)(40,45)(41,43)(44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=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, 9949796);