#8091729

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

4597 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)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24)(25,26,30,28)(27,33,34,31)(29,32,37,36)(35,39,40,38);(1,2)(3,6)(4,7)(5,8)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(10,15)(13,22)(16,23)(20,24)(27,34)(31,33)(35,40)(38,39);(1,3,5,4)(2,6,8,7)(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,28,30,26)(27,31,34,33)(29,32,37,36)(35,39,40,38);(1,3,5,4)(2,6,8,7)(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,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, 7998925);