#7945636

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7852832);