#7987473

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

4211 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)(9,10,14,16)(11,17,12,15)(13,18,21,23)(19,24,20,22)(25,26,32,30)(27,35,29,33)(28,34,31,36)(37,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,16,17)(13,20,21,19)(18,24,23,22)(25,27,32,29)(26,33,30,35)(28,38,31,37)(34,40,36,39);(10,16)(11,12)(18,23)(19,20)(25,28,32,31)(26,34,30,36)(27,37,29,38)(33,39,35,40);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,24)(21,23)(25,29,32,27)(26,35,30,33)(28,38,31,37)(34,40,36,39);(1,3,4,7)(2,5,6,8)(9,12,14,11)(10,17,16,15)(13,19,21,20)(18,22,23,24)(25,30,32,26)(27,35,29,33)(28,34,31,36)(37,39,38,40);(9,13,14,21)(10,18,16,23)(11,19,12,20)(15,22,17,24)(25,31,32,28)(26,36,30,34)(27,38,29,37)(33,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=hi,e^2=g,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, 7894669);