#7644317

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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)(11,12)(14,22)(16,19)(17,21)(25,26,31,34)(27,35,30,32)(28,38,33,29)(36,40,39,37);(1,2)(3,5)(4,6)(7,8)(9,10,15,13)(11,16,12,19)(14,17,22,21)(18,23,20,24)(25,27,31,30)(26,32,34,35)(28,39,33,36)(29,37,38,40);(9,11,15,12)(10,16,13,19)(14,18,22,20)(17,23,21,24)(25,28,31,33)(26,29,34,38)(27,36,30,39)(32,40,35,37);(9,12,15,11)(10,16,13,19)(14,20,22,18)(17,23,21,24)(25,29)(26,33)(27,37)(28,34)(30,40)(31,38)(32,36)(35,39);(9,13,15,10)(11,16,12,19)(14,21,22,17)(18,23,20,24);(1,3,4,7)(2,5,6,8)(9,14)(10,17)(11,18)(12,20)(13,21)(15,22)(16,23)(19,24)(25,30,31,27)(26,32,34,35)(28,39,33,36)(29,40,38,37)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551513);