#9206413

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3876 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)(3,8)(4,9)(5,14)(6,7)(10,16)(11,15)(12,13)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(2,7)(4,13)(8,15)(10,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,25)(26,27)(31,32)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(2,7)(3,11)(9,12)(10,14)(17,20,22,29)(18,24,26,32)(19,28,30,21)(23,25,27,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,32)(27,31)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,16,15)(17,21,22,28)(18,25,26,31)(19,29,30,20)(23,24,27,32)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,18)(19,23)(20,24)(21,31)(22,26)(25,28)(27,30)(29,32)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=gi,f^2=h,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, 9113609);