#8096292

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	#52307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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)(9,10,14,18)(11,17,15,12)(13,21,23,16)(19,20,22,24);(2,5)(3,7)(9,11)(10,12)(13,22)(14,15)(16,24)(17,18)(19,23)(20,21)(26,30)(31,37)(32,38)(39,40);(1,2,4,5)(3,6,7,8)(9,12)(10,15)(11,18)(13,20)(14,17)(16,22)(19,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13,14,23)(10,16,18,21)(11,19,15,22)(12,20,17,24);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,22)(23,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,11)(10,17)(12,18)(13,19)(14,15)(16,24)(20,21)(22,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=g,e^2=hi,f^2=hi,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, 8003488);