#7743459

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,7,8,5)(9,10,15,17)(11,16,14,12)(13,21,23,18)(19,20,22,24)(26,31)(27,28)(33,37)(35,36);(1,3,4,8)(2,5,6,7)(9,11,15,14)(10,16,17,12)(13,19,23,22)(18,24,21,20)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,39)(36,40);(10,17)(12,16)(18,21)(20,24)(25,27,30,28)(26,32,31,34)(29,35,38,36)(33,39,37,40);(9,12)(10,11)(13,20)(14,17)(15,16)(18,19)(21,22)(23,24)(25,28,30,27)(26,32,31,34)(29,36,38,35)(33,39,37,40);(9,13,15,23)(10,18,17,21)(11,19,14,22)(12,20,16,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,39)(37,38);(9,14,15,11)(10,12,17,16)(13,22,23,19)(18,20,21,24)(25,29,30,38)(26,33,31,37)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

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, 7650655);