#7895479

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×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4269 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)(9,10,15,17)(11,18,14,16)(12,13,22,24)(19,20,21,23)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,24,23);(9,12)(10,13)(11,19)(14,21)(15,22)(16,23)(17,24)(18,20)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(9,13)(10,12)(11,20)(14,23)(15,24)(16,21)(17,22)(18,19)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,12)(10,13)(11,21)(14,19)(15,22)(16,20)(17,24)(18,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,14,15,11)(10,16,17,18)(12,21,22,19)(13,23,24,20)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,f^2=hi,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, 7802675);