#9210285

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9117481);