#7866420

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 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)(9,10,15,12)(11,13,18,16)(14,17,23,20)(19,21,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11,15,18)(10,16,12,13)(14,19,23,24)(17,22,20,21)(26,31)(27,35)(33,36)(34,40);(10,12)(11,18)(17,20)(19,24)(25,27,30,35)(26,32,31,37)(28,34,38,40)(29,36,39,33);(9,12,15,10)(11,13,18,16)(14,20,23,17)(19,21,24,22);(9,13)(10,11)(12,18)(14,22)(15,16)(17,24)(19,20)(21,23)(25,28)(26,33)(27,34)(29,32)(30,38)(31,36)(35,40)(37,39);(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=h,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, 7773616);