#7711822

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4328 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)(13,21)(16,20)(18,24)(22,23)(25,26)(27,31)(28,38)(29,39)(30,34)(32,36)(33,40)(35,37);(1,2)(3,5)(4,6)(7,8)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(25,27,30,37)(26,31,34,35)(28,29,36,40)(32,33,38,39);(10,12)(15,17)(16,20)(22,23)(25,26,30,34)(27,35,37,31)(28,38,36,32)(29,33,40,39);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,38)(39,40);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,36)(26,32,34,38)(27,29,37,40)(31,33,35,39);(1,3,4,7)(2,5,6,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,36)(28,37)(30,40)(31,38)(32,35)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=i,f^2=g,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, 7619018);