#7944504

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gi,e^2=gh,f^2=gh,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, 7851700);