#7656719

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

4521 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)(11,17)(12,16)(13,18)(14,19)(15,22)(20,21)(23,24)(26,32)(28,34)(33,35)(36,38);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,26,31,32)(27,35,29,33)(28,30,34,40)(36,39,38,37);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,19)(18,24)(25,27,31,29)(26,33,32,35)(28,36,34,38)(30,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,14,18)(10,16,19,13)(11,20,22,24)(15,23,17,21)(25,28,31,34)(26,30,32,40)(27,36,29,38)(33,37,35,39);(1,4,5,3)(2,8,6,7)(9,11)(10,17)(12,20)(13,23)(14,22)(15,19)(16,21)(18,24)(25,29,31,27)(26,33,32,35)(28,38,34,36)(30,37,40,39);(9,13,14,16)(10,18,19,12)(11,21,22,23)(15,24,17,20)(25,30)(26,34)(27,37)(28,32)(29,39)(31,40)(33,38)(35,36)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563915);