#8097009

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	C₂×#548	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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)(3,5,8,7)(10,15)(11,17)(18,23)(19,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,17)(13,18)(14,15)(19,21)(20,24)(22,23)(27,33)(31,34)(35,39)(38,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,21,23,20);(9,12,14,16)(10,17,15,11)(13,21,22,20)(18,19,23,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,13,14,22)(10,18,15,23)(11,19,17,24)(12,20,16,21)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004205);