#7984023

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

4520 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,f^2=gi,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, 7891219);