#8075375

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×#562	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=h,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, 7982571);