#7644364

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4374 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551560);