#7883008

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4289 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)(9,10)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(25,26,32,38)(27,33,29,35)(28,36,30,34)(31,37,40,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,27,32,29)(26,33,38,35)(28,39,30,37)(31,34,40,36);(10,17)(12,20)(16,21)(19,24)(25,28,32,30)(26,34,38,36)(27,37,29,39)(31,35,40,33);(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,29,32,27)(26,35,38,33)(28,37,30,39)(31,36,40,34);(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,30,32,28)(26,36,38,34)(27,37,29,39)(31,35,40,33);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,31,32,40)(26,37,38,39)(27,34,29,36)(28,35,30,33)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7790204);