#8187009

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	#52963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094205);