#8075287

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×#560	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982483);