#9947386

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	#47895	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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)(10,14)(11,12)(15,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,28)(27,32)(29,31)(33,34)(35,36)(37,40)(38,46)(39,41)(42,47)(43,48)(44,45);(2,6)(3,9)(8,11)(10,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,39,37)(34,40,41,36)(38,43,47,45)(42,48,46,44);(18,19)(23,25)(24,26)(31,32)(34,41)(36,40)(42,46)(44,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36,39,40)(34,35,41,37)(38,44,47,48)(42,43,46,45);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,35)(34,40,41,36)(38,43,47,45)(42,44,46,48);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38,39,47)(34,42,41,46)(35,43,37,45)(36,44,40,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,e^2=h,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, 9854582);