#7782699

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

4278 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)(11,19)(12,16)(13,14)(15,17)(18,24)(20,23)(21,22);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,17,19)(13,20,24,22)(14,21,18,23)(25,26,31,30)(27,35,37,32)(28,33,36,38)(29,34,40,39);(10,17)(11,12)(14,18)(20,22)(25,27,31,37)(26,32,30,35)(28,29,36,40)(33,39,38,34);(9,12,15,11)(10,16,17,19)(13,20,24,22)(14,23,18,21)(25,28)(26,33)(27,29)(30,38)(31,36)(32,39)(34,35)(37,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,20)(12,22)(14,17)(15,24)(16,23)(19,21)(25,29)(26,34)(27,36)(28,37)(30,39)(31,40)(32,33)(35,38);(9,14,15,18)(10,13,17,24)(11,21,12,23)(16,20,19,22)(25,30,31,26)(27,32,37,35)(28,38,36,33)(29,39,40,34)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7689895);