#7987214

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

4187 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)(3,5,7,8)(9,10)(11,20)(12,14)(13,24)(15,17)(16,21)(18,23)(19,22)(25,26,30,29)(27,31,35,33)(28,36,37,32)(34,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,21)(10,16,17,20)(12,19,23,24)(13,18,22,14)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(10,17)(11,21)(12,23)(13,22)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(9,12)(10,18)(11,19)(13,20)(14,17)(15,23)(16,22)(21,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,5,6,8)(9,13)(10,19)(11,18)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,14,15,18)(10,12,17,23)(11,22,21,13)(16,24,20,19)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=hi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7894410);