#8550922

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=h,e^2=gh,f^2=gi,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, 8458118);