#7712455

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	#37277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4375 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)(9,10,15,12)(11,16,19,20)(13,23,22,17)(14,24,21,18)(26,31)(29,39)(32,38)(33,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,24)(18,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(10,12)(11,19)(13,22)(18,24)(26,31)(28,37)(32,38)(34,35);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,20,19,16)(13,23,22,17)(14,24,21,18)(25,27,30,36)(26,32,31,38)(28,35,37,34)(29,33,39,40);(9,13,15,22)(10,17,12,23)(11,21,19,14)(16,24,20,18)(25,28)(26,33)(27,35)(29,38)(30,37)(31,40)(32,39)(34,36);(9,14,15,21)(10,18,12,24)(11,22,19,13)(16,23,20,17)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,38,37,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=h,f^2=i,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, 7619651);