#9937776

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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)(10,15)(11,12)(14,16)(17,18)(19,23)(20,26)(21,29)(22,24)(25,28)(27,32)(30,31)(33,34,39,41)(35,45,46,40)(36,37,43,42)(38,48,47,44);(2,7)(3,9)(10,13)(11,16)(17,19,22,28)(18,23,24,25)(20,29,31,27)(21,30,32,26)(33,35)(34,40)(36,47)(37,48)(38,43)(39,46)(41,45)(42,44);(2,7)(3,9)(10,13)(11,16)(18,24)(19,28)(26,30)(27,29)(34,41)(35,46)(36,43)(44,48);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31)(33,36,39,43)(34,42,41,37)(35,47,46,38)(40,44,45,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,20)(18,26)(19,29)(21,23)(22,31)(24,30)(25,32)(27,28)(33,37,39,42)(34,43,41,36)(35,44,46,48)(38,40,47,45);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,21,22,32)(18,27,24,29)(19,30,28,26)(20,25,31,23)(33,38,39,47)(34,44,41,48)(35,43,46,36)(37,40,42,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,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=eghi,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, 9844972);