#7876943

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7784139);