#9947454

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	#37529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,28)(20,21)(22,32)(23,27)(24,30)(25,31)(26,29)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,19)(18,24)(20,22)(21,26)(23,30)(25,32)(27,28)(29,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(17,20,23,31)(18,25,27,21)(19,22,30,29)(24,32,28,26)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,21)(19,29)(22,30)(23,31)(24,32)(25,27)(26,28)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,21)(18,20)(19,26)(22,24)(23,25)(27,31)(28,29)(30,32)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22,23,29)(18,26,27,32)(19,20,30,31)(21,28,25,24)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=i,e^2=gh,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, 9854650);