#7986196

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,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, 7893392);