#7961804

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

4303 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)(9,10,15,17)(11,16,14,18)(12,13,21,24)(19,20,22,23)(25,26,30,32)(27,29,34,31)(28,36,38,33)(35,37,40,39);(1,2,5,6)(3,7,4,8)(9,11,15,14)(10,16,17,18)(12,19,21,22)(13,20,24,23)(25,27,30,34)(26,31,32,29)(28,35,38,40)(33,37,36,39);(10,17)(12,21)(16,18)(19,22)(26,32)(27,34)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,22)(15,21)(16,20)(17,24)(18,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,38)(39,40);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,20)(14,23)(15,24)(16,19)(17,21)(18,22)(25,28)(26,33)(27,35)(29,39)(30,38)(31,37)(32,36)(34,40);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,18,17,16)(12,22,21,19)(13,23,24,20)(25,29,30,31)(26,34,32,27)(28,37,38,39)(33,35,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=gi,f^2=ghi,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, 7869000);