#8550992

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458188);