#7709354

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	#59212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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)(11,18)(13,15)(19,20)(21,23)(25,26,32,31)(27,28,34,30)(29,33,39,38)(35,36,40,37);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,21)(20,23)(22,24)(25,27)(26,30)(28,31)(29,35)(32,34)(33,37)(36,38)(39,40);(9,11,14,18)(10,15,17,13)(12,20,22,19)(16,21,24,23)(25,28,32,30)(26,27,31,34)(29,37,39,36)(33,40,38,35);(9,12,14,22)(10,16,17,24)(11,19,18,20)(13,21,15,23)(25,29,32,39)(26,33,31,38)(27,35,34,40)(28,36,30,37);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,30,32,28)(26,34,31,27)(29,37,39,36)(33,40,38,35);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,21)(14,15)(16,19)(17,18)(20,24)(22,23)(25,31,32,26)(27,28,34,30)(29,38,39,33)(35,36,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,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, 7616550);