#9221186

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9128382);