#9947296

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

3882 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18)(19,21)(20,27)(22,25)(23,30)(24,31)(26,32)(28,29)(33,34)(35,36)(37,41)(38,46)(39,40)(42,47)(43,48)(44,45);(2,6)(3,9)(8,14)(10,11)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(34,40)(36,41)(42,46)(44,48);(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,18)(19,21)(20,23)(22,25)(24,31)(26,29)(27,30)(28,32)(33,37,39,35)(34,41,40,36)(38,43,47,45)(42,44,46,48);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=g,f^2=g,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, 9854492);