#8091724

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

4466 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=ghi,f^2=gi,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, 7998920);