#8947063

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	#41976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8854259);