#9206462

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	#51967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(19,20)(21,29)(22,25)(23,27)(24,31)(26,30)(28,32)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(34,39)(35,40)(37,45)(44,48);(4,12)(9,13)(11,14)(15,16)(17,18,22,25)(19,23,27,20)(21,24,31,29)(26,30,32,28)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,4,7,12)(2,9,6,13)(3,11,5,14)(8,15,10,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,29)(27,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=gi,f^2=gh,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, 9113658);