#7920126

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7827322);