#7987967

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895163);