#8091749

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7998945);