#7883084

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	#2407	C₂²
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790280);