#7720250

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	#34609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^2=h,f^2=hi,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, 7627446);