#7644750

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	#37444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7551946);