#7719919

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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)(12,20)(16,21)(19,23)(22,24)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,5)(4,6)(7,8)(10,14)(11,17)(16,21)(19,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(1,3,4,7)(2,5,6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,27)(26,32,34,31)(29,35,37,36)(33,38,40,39);(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627115);