#7917614

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

4373 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)(11,16,20,23)(12,22,18,13)(14,19,21,24)(25,26,30,34)(27,31,36,39)(28,32,29,33)(35,38,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,21)(13,19)(14,18)(15,20)(17,23)(22,24)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,38)(34,39);(1,3,5,4)(2,7,6,8)(10,17)(11,20)(12,18)(19,24)(25,28,30,29)(26,32,34,33)(27,35,36,37)(31,38,39,40);(9,12,15,18)(10,13,17,22)(11,21,20,14)(16,19,23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,17)(14,16)(15,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,38)(33,40)(34,39);(9,14,15,21)(10,19,17,24)(11,18,20,12)(13,23,22,16)(25,29,30,28)(26,33,34,32)(27,35,36,37)(31,38,39,40)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,e^2=g,f^2=i,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, 7824810);