#7903108

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
prime radical	#36083	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=ghi,e^2=h,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, 7810304);