#7709300

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	#59213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,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, 7616496);