#7709870

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	#45616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4387 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617066);